-
question_answer1)
\[\frac{d}{dx}(\log \tan x)=\] [MNR 1986]
A)
\[2\sec 2x\] done
clear
B)
\[2\,\text{cosec }2x\] done
clear
C)
\[\sec 2x\] done
clear
D)
\[\text{cosec}\,2x\] done
clear
View Solution play_arrow
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question_answer2)
\[\frac{d}{dx}\log (\log x)\]= [IIT 1985]
A)
\[\frac{x}{\log x}\] done
clear
B)
\[\frac{\log x}{x}\] done
clear
C)
\[{{(x\log x)}^{-1}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer3)
\[\frac{d}{dx}{{\left( \sqrt{x}+\frac{1}{\sqrt{x}} \right)}^{2}}=\] [AI CBSE 1980]
A)
\[1-\frac{1}{{{x}^{2}}}\] done
clear
B)
\[1+\frac{1}{{{x}^{2}}}\] done
clear
C)
\[1-\frac{1}{2x}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer4)
If \[y=x+\frac{1}{x}\], then
A)
\[{{x}^{2}}\frac{dy}{dx}+xy=0\] done
clear
B)
\[{{x}^{2}}\frac{dy}{dx}+xy+2=0\] done
clear
C)
\[{{x}^{2}}\frac{dy}{dx}-xy+2=0\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer5)
\[\frac{d}{dx}\left( \frac{1}{{{x}^{4}}\sec x} \right)=\]
A)
\[\frac{x\sin x+4\cos x}{{{x}^{5}}}\] done
clear
B)
\[\frac{-(x\sin x+4\cos x)}{{{x}^{5}}}\] done
clear
C)
\[\frac{4\cos x-x\sin x}{{{x}^{5}}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer6)
If \[y=1+x+\frac{{{x}^{2}}}{2\,!}+\frac{{{x}^{3}}}{3\,!}+.....+\frac{{{x}^{n}}}{n\,!}\], then \[\frac{dy}{dx}=\]
A)
y done
clear
B)
\[y+\frac{{{x}^{n}}}{n!}\] done
clear
C)
\[y-\frac{{{x}^{n}}}{n!}\] done
clear
D)
\[y-1-\frac{{{x}^{n}}}{n!}\] done
clear
View Solution play_arrow
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question_answer7)
If \[y=1+x+\frac{{{x}^{2}}}{2!}+\frac{{{x}^{3}}}{3!}+.....\infty ,\]then \[\frac{dy}{dx}=\] [Karnataka CET 1999]
A)
y done
clear
B)
\[y-1\] done
clear
C)
\[y+1\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer8)
If \[y=\frac{1}{a-z},\]then \[\frac{dz}{dy}=\]
A)
\[{{(z-a)}^{2}}\] done
clear
B)
\[-{{(z-a)}^{2}}\] done
clear
C)
\[{{(z+a)}^{2}}\] done
clear
D)
\[-{{(z+a)}^{2}}\] done
clear
View Solution play_arrow
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question_answer9)
If \[y=x\sin x,\]then
A)
\[\frac{1}{y}\frac{dy}{dx}=\frac{1}{x}+\cot x\] done
clear
B)
\[\frac{dy}{dx}=\frac{1}{x}+\cot x\] done
clear
C)
\[\frac{1}{y}\frac{dy}{dx}=\frac{1}{x}-\cot x\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer10)
\[\frac{d}{dx}({{x}^{2}}{{e}^{x}}\sin x)=\]
A)
\[x\,{{e}^{x}}(2\sin x+x\sin x+x\cos x)\] done
clear
B)
\[x\,{{e}^{x}}(2\sin x+x\sin x-\cos x)\] done
clear
C)
\[x\,{{e}^{x}}(2\sin x+x\sin x+\cos x)\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer11)
\[\frac{d}{dx}\left( {{\tan }^{-1}}\frac{\cos x}{1+\sin x} \right)=\] [AISSE 1984, 85; MNR 1983; RPET 1997]
A)
\[-\frac{1}{2}\] done
clear
B)
\[\frac{1}{2}\] done
clear
C)
\[-1\] done
clear
D)
1 done
clear
View Solution play_arrow
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question_answer12)
\[\frac{d}{dx}[\cos {{(1-{{x}^{2}})}^{2}}]\]= [AISSE 1981; AI CBSE 1979]
A)
\[-2x(1-{{x}^{2}})\sin {{(1-{{x}^{2}})}^{2}}\] done
clear
B)
\[-4x(1-{{x}^{2}})\sin {{(1-{{x}^{2}})}^{2}}\] done
clear
C)
\[4x(1-{{x}^{2}})\sin {{(1-{{x}^{2}})}^{2}}\] done
clear
D)
\[-2(1-{{x}^{2}})\sin {{(1-{{x}^{2}})}^{2}}\] done
clear
View Solution play_arrow
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question_answer13)
\[\frac{d}{dx}\left( {{x}^{2}}\sin \frac{1}{x} \right)=\]
A)
\[\cos \,\left( \frac{1}{x} \right)+2x\sin \left( \frac{1}{x} \right)\] done
clear
B)
\[2x\sin \left( \frac{1}{x} \right)-\cos \left( \frac{1}{x} \right)\] done
clear
C)
\[\cos \left( \frac{1}{x} \right)-2x\sin \left( \frac{1}{x} \right)\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer14)
If \[y=\cos (\sin {{x}^{2}}),\]then at \[x=\sqrt{\frac{\pi }{2}},\frac{dy}{dx}\]=
A)
? 2 done
clear
B)
2 done
clear
C)
\[-2\sqrt{\frac{\pi }{2}}\] done
clear
D)
0 done
clear
View Solution play_arrow
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question_answer15)
If \[y={{\sin }^{-1}}(x\sqrt{1-x}+\sqrt{x}\sqrt{1-{{x}^{2}})},\]then \[\frac{dy}{dx}=\] [Roorkee 1981; MP PET 2004]
A)
\[\frac{-2x}{\sqrt{1-{{x}^{2}}}}+\frac{1}{2\sqrt{x-{{x}^{2}}}}\] done
clear
B)
\[\frac{-1}{\sqrt{1-{{x}^{2}}}}-\frac{1}{2\sqrt{x-{{x}^{2}}}}\] done
clear
C)
\[\frac{1}{\sqrt{1-{{x}^{2}}}}+\frac{1}{2\sqrt{x-{{x}^{2}}}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer16)
\[\frac{d}{dx}\log |x|\text{ }=......,(x\ne 0)\]
A)
\[\frac{1}{x}\] done
clear
B)
\[-\frac{1}{x}\] done
clear
C)
x done
clear
D)
\[-x\] done
clear
View Solution play_arrow
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question_answer17)
If \[y=a\sin x+b\cos x,\]then \[{{y}^{2}}+{{\left( \frac{dy}{dx} \right)}^{2}}\]is a
A)
Function of x done
clear
B)
Function of y done
clear
C)
Function of x and y done
clear
D)
Constant done
clear
View Solution play_arrow
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question_answer18)
\[f(x)={{x}^{2}}-3x\], then the points at which \[f(x)=f'(x)\]are
A)
1, 3 done
clear
B)
1, ? 3 done
clear
C)
? 1, 3 done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer19)
If \[f(x)=mx+c,f(0)=f'(0)=1\]then \[f(2)=\]
A)
1 done
clear
B)
2 done
clear
C)
3 done
clear
D)
? 3 done
clear
View Solution play_arrow
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question_answer20)
If \[y=3{{x}^{5}}+4{{x}^{4}}+2x+3\], then
A)
\[{{y}_{4}}=0\] done
clear
B)
\[{{y}_{5}}=0\] done
clear
C)
\[{{y}_{6}}=0\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer21)
If \[y=x\text{ }\left[ \left( \cos \frac{x}{2}+\sin \frac{x}{2} \right)\text{ }\left( \cos \frac{x}{2}-\sin \frac{x}{2} \right)+\sin x \right]+\frac{1}{2\sqrt{x}}\], then \[\frac{dy}{dx}=\]
A)
\[(1+x)\cos x+(1-x)\sin x-\frac{1}{4x\sqrt{x}}\] done
clear
B)
\[(1-x)\cos x+(1+x)\sin x+\frac{1}{4x\sqrt{x}}\] done
clear
C)
\[(1+x)\cos x+(1+x)\sin x-\frac{1}{4x\sqrt{x}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer22)
The differential coefficient of \[{{a}^{x}}+\log x.\sin x\]is
A)
\[{{a}^{x}}{{\log }_{e}}a+\frac{\sin x}{x}+\log x.\cos x\] done
clear
B)
\[{{a}^{x}}+\frac{\sin x}{x}+\cos x.\log x\] done
clear
C)
\[{{a}^{x}}\log a+\frac{\cos x}{x}+\sin x.\log x.\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer23)
\[\frac{d}{dx}{{\tan }^{-1}}\left( \frac{ax-b}{bx+a} \right)=\]
A)
\[\frac{1}{1+{{x}^{2}}}-\frac{{{a}^{2}}}{{{a}^{2}}+{{b}^{2}}}\] done
clear
B)
\[\frac{-1}{1+{{x}^{2}}}-\frac{{{a}^{2}}}{{{a}^{2}}+{{b}^{2}}}\] done
clear
C)
\[\frac{1}{1+{{x}^{2}}}+\frac{{{a}^{2}}}{{{a}^{2}}+{{b}^{2}}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer24)
\[\frac{d}{dx}\left( {{\tan }^{-1}}\sqrt{\frac{1+\cos \frac{x}{2}}{1-\cos \frac{x}{2}}} \right)\]is equal to [MP PET 2004]
A)
\[-\frac{1}{4}\] done
clear
B)
\[\frac{1}{2}\] done
clear
C)
\[-\frac{1}{2}\] done
clear
D)
\[\frac{1}{4}\] done
clear
View Solution play_arrow
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question_answer25)
\[\frac{d}{dx}\sqrt{\frac{1-\sin 2x}{1+\sin 2x}}=\] [AISSE 1985; DSSE 1986]
A)
\[{{\sec }^{2}}x\] done
clear
B)
\[-{{\sec }^{2}}\left( \frac{\pi }{4}-x \right)\] done
clear
C)
\[{{\sec }^{2}}\left( \frac{\pi }{4}+x \right)\] done
clear
D)
\[{{\sec }^{2}}\left( \frac{\pi }{4}-x \right)\] done
clear
View Solution play_arrow
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question_answer26)
If \[y=\sqrt{(1-x)(1+x)}\], then
A)
\[(1-{{x}^{2}})\frac{dy}{dx}-xy=0\] done
clear
B)
\[(1-{{x}^{2}})\frac{dy}{dx}+xy=0\] done
clear
C)
\[(1-{{x}^{2}})\frac{dy}{dx}-2xy=0\] done
clear
D)
\[(1-{{x}^{2}})\frac{dy}{dx}+2xy=0\] done
clear
View Solution play_arrow
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question_answer27)
\[\frac{d}{dx}\left( \frac{{{\cot }^{2}}x-1}{{{\cot }^{2}}x+1} \right)=\]
A)
\[-\sin 2x\] done
clear
B)
\[2\sin 2x\] done
clear
C)
\[2\cos 2x\] done
clear
D)
\[-2\sin 2x\] done
clear
View Solution play_arrow
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question_answer28)
If \[f(x)=x{{\tan }^{-1}}x\], then \[f'(1)\]= [IIT 1979]
A)
\[1+\frac{\pi }{4}\] done
clear
B)
\[\frac{1}{2}+\frac{\pi }{4}\] done
clear
C)
\[\frac{1}{2}-\frac{\pi }{4}\] done
clear
D)
2 done
clear
View Solution play_arrow
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question_answer29)
If \[y={{\log }_{10}}x+{{\log }_{x}}10+{{\log }_{x}}x+{{\log }_{10}}10,\]then \[\frac{dy}{dx}=\]
A)
\[\frac{1}{x{{\log }_{e}}10}-\frac{{{\log }_{e}}10}{x{{({{\log }_{e}}x)}^{2}}}\] done
clear
B)
\[\frac{1}{x{{\log }_{e}}10}-\frac{1}{x{{\log }_{10}}e}\] done
clear
C)
\[\frac{1}{x{{\log }_{e}}10}-\frac{{{\log }_{e}}10}{x{{({{\log }_{e}}x)}^{2}}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer30)
If \[y=b\cos \log {{\left( \frac{x}{n} \right)}^{n}}\], then \[\frac{dy}{dx}=\]
A)
\[-n\,\,b\sin \log {{\left( \frac{x}{n} \right)}^{n}}\] done
clear
B)
\[n\,\,b\sin \log {{\left( \frac{x}{n} \right)}^{n}}\] done
clear
C)
\[-n\,b\sin \log {{\left( \frac{x}{n} \right)}^{n}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer31)
\[\frac{d}{dx}[{{\sin }^{n}}x\cos \,nx]=\]
A)
\[n{{\sin }^{n-1}}x\cos (n+1)x\] done
clear
B)
\[n{{\sin }^{n-1}}x\cos \,nx\] done
clear
C)
\[n{{\sin }^{n-1}}x\cos (n-1)x\] done
clear
D)
\[n{{\sin }^{n-1}}x\sin (n+1)x\] done
clear
View Solution play_arrow
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question_answer32)
If \[f(x)={{\log }_{x}}(\log x),\]then \[f'(x)\]at \[x=e\]is [IIT 1985; RPET 2000; MP PET 2000; Karnataka CET 2002; Pb. CET 2002]
A)
e done
clear
B)
\[\frac{1}{e}\] done
clear
C)
1 done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer33)
If \[y=\log {{\left( \frac{1+x}{1-x} \right)}^{1/4}}-\frac{1}{2}{{\tan }^{-1}}x,\]then \[\frac{dy}{dx}=\]
A)
\[\frac{{{x}^{2}}}{1-{{x}^{4}}}\] done
clear
B)
\[\frac{2{{x}^{2}}}{1-{{x}^{4}}}\] done
clear
C)
\[\frac{{{x}^{2}}}{2\,\,(1-{{x}^{4}})}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer34)
If \[y={{\tan }^{-1}}\frac{4x}{1+5{{x}^{2}}}+{{\tan }^{-1}}\frac{2+3x}{3-2x}\], then \[\frac{dy}{dx}=\]
A)
\[\frac{1}{1+25{{x}^{2}}}+\frac{2}{1+{{x}^{2}}}\] done
clear
B)
\[\frac{5}{1+25{{x}^{2}}}+\frac{2}{1+{{x}^{2}}}\] done
clear
C)
\[\frac{5}{1+25{{x}^{2}}}\] done
clear
D)
\[\frac{1}{1+25{{x}^{2}}}\] done
clear
View Solution play_arrow
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question_answer35)
\[\frac{d}{dx}{{\log }_{7}}({{\log }_{7}}x)\]=
A)
\[\frac{1}{x{{\log }_{e}}x}\] done
clear
B)
\[\frac{{{\log }_{e}}7}{x{{\log }_{e}}x}\] done
clear
C)
\[\frac{{{\log }_{7}}e}{x{{\log }_{e}}x}\] done
clear
D)
\[\frac{{{\log }_{7}}e}{x{{\log }_{7}}x}\] done
clear
View Solution play_arrow
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question_answer36)
If \[f(x)=\sqrt{1+{{\cos }^{2}}({{x}^{2}})}\], then \[f'\left( \frac{\sqrt{\pi }}{2} \right)\] is [Orissa JEE 2004]
A)
\[\sqrt{\pi }/6\] done
clear
B)
\[-\,\sqrt{(\pi /6)}\] done
clear
C)
\[1/\sqrt{6}\] done
clear
D)
\[\pi /\sqrt{6}\] done
clear
View Solution play_arrow
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question_answer37)
If \[{{x}^{2/3}}+{{y}^{2/3}}={{a}^{2/3}}\], then \[\frac{dy}{dx}=\]
A)
\[{{\left( \frac{y}{x} \right)}^{1/3}}\] done
clear
B)
\[-{{\left( \frac{y}{x} \right)}^{1/3}}\] done
clear
C)
\[{{\left( \frac{x}{y} \right)}^{1/3}}\] done
clear
D)
\[-{{\left( \frac{x}{y} \right)}^{1/3}}\] done
clear
View Solution play_arrow
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question_answer38)
\[\frac{d}{dx}[(1+{{x}^{2}}){{\tan }^{-1}}x]=\]
A)
\[x{{\tan }^{-1}}x\] done
clear
B)
\[2{{\tan }^{-1}}x\] done
clear
C)
\[2x{{\tan }^{-1}}x+1\] done
clear
D)
\[x{{\tan }^{-1}}x+1\] done
clear
View Solution play_arrow
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question_answer39)
If \[y=\log \frac{1+\sqrt{x}}{1-\sqrt{x}},\]then \[\frac{dy}{dx}=\]
A)
\[\frac{\sqrt{x}}{1-x}\] done
clear
B)
\[\frac{1}{\sqrt{x}(1-x)}\] done
clear
C)
\[\frac{\sqrt{x}}{1+x}\] done
clear
D)
\[\frac{1}{\sqrt{x}(1+x)}\] done
clear
View Solution play_arrow
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question_answer40)
\[\frac{d}{dx}{{e}^{x+3\log x}}=\]
A)
\[{{e}^{x}}.{{x}^{2}}(x+3)\] done
clear
B)
\[{{e}^{x}}.x(x+3)\] done
clear
C)
\[{{e}^{x}}+\frac{3}{x}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer41)
\[\frac{d}{dx}\sqrt{\frac{1+\cos 2x}{1-\cos 2x}}=\]
A)
\[{{\sec }^{2}}x\] done
clear
B)
\[-\text{cose}{{\text{c}}^{2}}x\] done
clear
C)
\[2\,{{\sec }^{2}}\frac{x}{2}\] done
clear
D)
\[-2\text{cose}{{\text{c}}^{2}}\frac{x}{2}\] done
clear
View Solution play_arrow
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question_answer42)
\[\frac{d}{dx}\log \tan \left( \frac{\pi }{4}+\frac{x}{2} \right)=\]
A)
\[\cos \text{ec}\,x\] done
clear
B)
\[-\cos \text{ec}\,x\] done
clear
C)
\[\sec x\] done
clear
D)
\[-\sec x\] done
clear
View Solution play_arrow
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question_answer43)
\[\frac{d}{dx}\log (\sqrt{x-a}+\sqrt{x-b})=\]
A)
\[\frac{1}{2[\sqrt{(x-a)}+\sqrt{(x-b)}]}\] done
clear
B)
\[\frac{1}{2\sqrt{(x-a)(x-b)}}\] done
clear
C)
\[\frac{1}{\sqrt{(x-a)(x-b)}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer44)
\[\frac{d}{dx}{{\tan }^{-1}}(\sec x+\tan x)=\] [AISSE 1985, 87; DSSE 1982, 84]
A)
1 done
clear
B)
1/2 done
clear
C)
\[\cos x\] done
clear
D)
\[\sec x\] done
clear
View Solution play_arrow
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question_answer45)
\[\frac{d}{dx}{{\cos }^{-1}}\sqrt{\cos x}=\]
A)
\[\frac{1}{2}\sqrt{1+\sec x}\] done
clear
B)
\[\sqrt{1+\sec x}\] done
clear
C)
\[-\frac{1}{2}\sqrt{1+\sec x}\] done
clear
D)
\[-\sqrt{1+\sec x}\] done
clear
View Solution play_arrow
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question_answer46)
\[\frac{d}{dx}({{e}^{x}}\log \sin 2x)=\] [AI CBSE 1985]
A)
\[{{e}^{x}}(\log \sin 2x+2\cot 2x)\] done
clear
B)
\[{{e}^{x}}(\log \cos 2x+2\cot 2x)\] done
clear
C)
\[{{e}^{x}}(\log \cos 2x+\cot 2x)\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer47)
\[\frac{d}{dx}{{\tan }^{-1}}\frac{4\sqrt{x}}{1-4x}=\]
A)
\[\frac{1}{\sqrt{x}(1+4x)}\] done
clear
B)
\[\frac{2}{\sqrt{x}(1+4x)}\] done
clear
C)
\[\frac{4}{\sqrt{x}(1+4x)}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer48)
If \[y=\sin [\cos (\sin x)],\]then \[dy/dx=\]
A)
\[-\cos [\cos (\sin x)]\sin (\cos x).\cos x\] done
clear
B)
\[-\cos [\cos (\sin x)]\sin (\sin x).\cos x\] done
clear
C)
\[\cos [\cos (\sin x)]\sin (\cos x).\cos x\] done
clear
D)
\[\cos [\cos (\sin x)]\sin (\sin x).\cos x\] done
clear
View Solution play_arrow
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question_answer49)
If \[y={{\sec }^{-1}}\left( \frac{\sqrt{x}+1}{\sqrt{x}-1} \right)+{{\sin }^{-1}}\left( \frac{\sqrt{x}-1}{\sqrt{x}+1} \right)\], then \[\frac{dy}{dx}=\] [UPSEAT 1999; AMU 2002; Kerala (Engg.) 2005]
A)
0 done
clear
B)
\[\frac{1}{\sqrt{x}+1}\] done
clear
C)
1 done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer50)
\[\frac{d}{dx}{{\sin }^{-1}}(3x-4{{x}^{3}})=\] [RPET 2003]
A)
\[\frac{3}{\sqrt{1-{{x}^{2}}}}\] done
clear
B)
\[\frac{-3}{\sqrt{1-{{x}^{2}}}}\] done
clear
C)
\[\frac{1}{\sqrt{1-{{x}^{2}}}}\] done
clear
D)
\[\frac{-1}{\sqrt{1-{{x}^{2}}}}\] done
clear
View Solution play_arrow
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question_answer51)
If \[y={{t}^{4/3}}-3{{t}^{-2/3}}\], then \[dy/dt\]=
A)
\[\frac{2{{t}^{2}}+3}{3{{t}^{5/3}}}\] done
clear
B)
\[\frac{2{{t}^{2}}+3}{{{t}^{5/3}}}\] done
clear
C)
\[\frac{2(2{{t}^{2}}+3)}{{{t}^{5/3}}}\] done
clear
D)
\[\frac{2(2{{t}^{2}}+3)}{3{{t}^{5/3}}}\] done
clear
View Solution play_arrow
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question_answer52)
If \[y={{x}^{2}}\log x+\frac{2}{\sqrt{x}},\] then \[\frac{dy}{dx}=\]
A)
\[x+2x\log x-\frac{1}{\sqrt{x}}\] done
clear
B)
\[x+2x\log x-\frac{1}{{{x}^{3/2}}}\] done
clear
C)
\[x+2x\log x-\frac{2}{{{x}^{3/2}}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer53)
\[\frac{d}{dx}\left( \frac{{{e}^{x}}}{1+{{x}^{2}}} \right)=\]
A)
\[\frac{{{e}^{x}}(1+x)}{{{(1+{{x}^{2}})}^{2}}}\] done
clear
B)
\[\frac{{{e}^{x}}{{(1-x)}^{2}}}{{{(1+{{x}^{2}})}^{2}}}\] done
clear
C)
\[\frac{{{e}^{x}}{{(1+x)}^{2}}}{(1+{{x}^{2}})}\] done
clear
D)
\[\frac{{{e}^{x}}{{(1-x)}^{2}}}{(1+{{x}^{2}})}\] done
clear
View Solution play_arrow
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question_answer54)
If \[y=\frac{\tan x+\cot x}{\tan x-\cot x},\]then \[\frac{dy}{dx}=\]
A)
\[2\tan 2x\sec 2x\] done
clear
B)
\[\tan 2x\sec 2x\] done
clear
C)
\[-\tan 2x\sec 2x\] done
clear
D)
\[-2\tan 2x\sec 2x\] done
clear
View Solution play_arrow
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question_answer55)
If \[A=\frac{{{2}^{x}}\cot x}{\sqrt{x}},\]then \[\frac{dA}{dx}=\]
A)
\[\frac{{{2}^{x-1}}\left\{ -2x\,\text{cos}\text{e}{{\text{c}}^{2}}x+\cot x.\log \left( \frac{{{4}^{x}}}{e} \right) \right\}}{{{x}^{3/2}}}\] done
clear
B)
\[\frac{{{2}^{x-1}}\left\{ -2x\cos \text{e}{{\text{c}}^{2}}x+\cot x.\log \left( \frac{{{4}^{x}}}{e} \right) \right\}}{x}\] done
clear
C)
\[\frac{2x\left\{ -2x\text{cose}{{\text{c}}^{2}}x+\cot x.\log \left( \frac{{{4}^{x}}}{e} \right) \right\}}{{{x}^{\text{3/2}}}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer56)
\[\frac{d}{dx}\left( \frac{\log x}{\sin x} \right)=\]
A)
\[\frac{\frac{\sin x}{x}-\log x.\cos x}{\sin x}\] done
clear
B)
\[\frac{\frac{\sin x}{x}-\log x.\cos x}{{{\sin }^{2}}x}\] done
clear
C)
\[\frac{\sin x-\log x.\cos x}{{{\sin }^{2}}x}\] done
clear
D)
\[\frac{\frac{\sin x}{x}-\log x}{{{\sin }^{2}}x}\] done
clear
View Solution play_arrow
-
question_answer57)
If \[y=\frac{\sqrt{{{x}^{2}}+1}+\sqrt{{{x}^{2}}-1}}{\sqrt{{{x}^{2}}+1}-\sqrt{{{x}^{2}}-1}}\], then \[\frac{dy}{dx}=\]
A)
\[2x+\frac{2{{x}^{3}}}{\sqrt{{{x}^{4}}-1}}\] done
clear
B)
\[2x+\frac{{{x}^{3}}}{\sqrt{{{x}^{4}}-1}}\] done
clear
C)
\[x+\frac{2{{x}^{3}}}{\sqrt{{{x}^{4}}-1}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer58)
If \[y=\frac{\sqrt{a+x}-\sqrt{a-x}}{\sqrt{a+x}+\sqrt{a-x}}\], then \[\frac{dy}{dx}=\] [AISSE 1986]
A)
\[\frac{ay}{x\sqrt{{{a}^{2}}-{{x}^{2}}}}\] done
clear
B)
\[\frac{ay}{\sqrt{{{a}^{2}}-{{x}^{2}}}}\] done
clear
C)
\[\frac{ay}{x\sqrt{{{x}^{2}}-{{a}^{2}}}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer59)
If \[y={{(x{{\cot }^{3}}x)}^{3/2}},\]then \[dy/dx=\]
A)
\[\frac{3}{2}{{(x{{\cot }^{3}}x)}^{1/2}}[{{\cot }^{3}}x-3x{{\cot }^{2}}x\cos \text{e}{{\text{c}}^{2}}x]\] done
clear
B)
\[\frac{3}{2}{{(x{{\cot }^{3}}x)}^{1/2}}[{{\cot }^{2}}x-3x{{\cot }^{2}}x\,\text{cose}{{\text{c}}^{2}}x]\] done
clear
C)
\[\frac{3}{2}{{(x{{\cot }^{3}}x)}^{1/3}}[{{\cot }^{3}}x-3x\,\text{cos}\text{e}{{\text{c}}^{2}}x]\] done
clear
D)
\[\frac{3}{2}{{(x{{\cot }^{3}}x)}^{3/2}}[{{\cot }^{3}}x-3x\,\text{cos}\text{e}{{\text{c}}^{2}}x]\] done
clear
View Solution play_arrow
-
question_answer60)
\[\frac{d}{dx}\{\cos (\sin {{x}^{2}})\}=\] [DSSE 1979]
A)
\[\sin (\sin {{x}^{2}}).\cos {{x}^{2}}.2x\] done
clear
B)
\[-\sin (\sin {{x}^{2}}).\cos {{x}^{2}}.2x\] done
clear
C)
\[-\sin (\sin {{x}^{2}}).{{\cos }^{2}}x.2x\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer61)
If \[y=\sin (\sqrt{\sin x+\cos x})\], then \[\frac{dy}{dx}=\] [DSSE 1987]
A)
\[\frac{1}{2}\frac{\cos \sqrt{\sin x+\cos x}}{\sqrt{\sin x+\cos x}}\] done
clear
B)
\[\frac{\cos \sqrt{\sin x+\cos x}}{\sqrt{\sin x+\cos x}}\] done
clear
C)
\[\frac{1}{2}\frac{\cos \sqrt{\sin x+\cos x}}{\sqrt{\sin x+\cos x}}.(\cos x-\sin x)\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer62)
If \[y=\sin \left( \frac{1+{{x}^{2}}}{1-{{x}^{2}}} \right)\], then \[\frac{dy}{dx}=\] [AISSE 1987]
A)
\[\frac{4x}{1-{{x}^{2}}}.\cos \left( \frac{1+{{x}^{2}}}{1-{{x}^{2}}} \right)\] done
clear
B)
\[\frac{x}{{{(1-{{x}^{2}})}^{2}}}.\cos \left( \frac{1+{{x}^{2}}}{1-{{x}^{2}}} \right)\] done
clear
C)
\[\frac{x}{(1-{{x}^{2}})}.\cos \left( \frac{1+{{x}^{2}}}{1-{{x}^{2}}} \right)\] done
clear
D)
\[\frac{4x}{{{(1-{{x}^{2}})}^{2}}}.\cos \left( \frac{1+{{x}^{2}}}{1-{{x}^{2}}} \right)\] done
clear
View Solution play_arrow
-
question_answer63)
If \[y=\sqrt{\frac{1+\tan x}{1-\tan x}}\], then \[\frac{dy}{dx}=\] [AISSE 1981, 83, 84, 85; DSSE 1985; AI CBSE 1981, 83]
A)
\[\frac{1}{2}\sqrt{\frac{1-\tan x}{1+\tan x}}.{{\sec }^{2}}\left( \frac{\pi }{4}+x \right)\] done
clear
B)
\[\sqrt{\frac{1-\tan x}{1+\tan x}}.{{\sec }^{2}}\left( \frac{\pi }{4}+x \right)\] done
clear
C)
\[\frac{1}{2}\sqrt{\frac{1-\tan x}{1+\tan x}}.\sec \left( \frac{\pi }{4}+x \right)\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer64)
\[\frac{d}{dx}{{({{x}^{2}}+\cos x)}^{4}}=\] [DSSE 1979]
A)
\[4({{x}^{2}}+\cos x)(2x-\sin x)\] done
clear
B)
\[4{{({{x}^{2}}-\cos x)}^{3}}(2x-\sin x)\] done
clear
C)
\[4{{({{x}^{2}}+\cos x)}^{3}}(2x-\sin x)\] done
clear
D)
\[4{{({{x}^{2}}+\cos x)}^{3}}(2x+\sin x)\] done
clear
View Solution play_arrow
-
question_answer65)
\[\frac{d}{dx}\sqrt{x\sin x}=\] [AISSE 1985]
A)
\[\frac{\sin x+x\cos x}{2\sqrt{x\sin x}}\] done
clear
B)
\[\frac{\sin x+x\cos x}{\sqrt{x\sin x}}\] done
clear
C)
\[\frac{x\sin x+\cos x}{\sqrt{2\sin x}}\] done
clear
D)
\[\frac{\sin x+x\cos x}{2\sqrt{x\sin x}}\] done
clear
View Solution play_arrow
-
question_answer66)
\[\frac{d}{dx}\sqrt{{{\sec }^{2}}x+\text{cose}{{\text{c}}^{2}}x}=\] [DSSE 1981]
A)
\[4\cos \text{ec 2}x.\cot 2x\] done
clear
B)
\[-4\cos \text{ec 2}x.\cot 2x\] done
clear
C)
\[-4\cos \text{ec }x.\cot 2x\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer67)
\[\frac{d}{dx}\left( \frac{\sec x+\tan x}{\sec x-\tan x} \right)=\] [DSSE 1979, 81; CBSE 1981]
A)
\[\frac{2\cos x}{{{(1-\sin x)}^{2}}}\] done
clear
B)
\[\frac{\cos x}{{{(1-\sin x)}^{2}}}\] done
clear
C)
\[\frac{2\cos x}{1-\sin x}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer68)
\[\frac{d}{dx}\left( {{x}^{3}}{{\tan }^{2}}\frac{x}{2} \right)\]= [AISSE 1979]
A)
\[{{x}^{3}}\tan \frac{x}{2}.{{\sec }^{2}}\frac{x}{2}+3x{{\tan }^{2}}\frac{x}{2}\] done
clear
B)
\[{{x}^{3}}\tan \frac{x}{2}.{{\sec }^{2}}\frac{x}{2}+3{{x}^{2}}{{\tan }^{2}}\frac{x}{2}\] done
clear
C)
\[{{x}^{3}}{{\tan }^{2}}\frac{x}{2}.{{\sec }^{2}}\frac{x}{2}+3{{x}^{2}}{{\tan }^{2}}\frac{x}{2}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer69)
If \[y={{\tan }^{-1}}\left( \frac{{{x}^{1/3}}+{{a}^{1/3}}}{1-{{x}^{1/3}}{{a}^{1/3}}} \right)\], then \[\frac{dy}{dx}=\] [DSSE 1986]
A)
\[\frac{1}{3{{x}^{2/3}}(1+{{x}^{2/3}})}\] done
clear
B)
\[\frac{1}{3{{x}^{2/3}}(1+{{x}^{2/3}})}\] done
clear
C)
\[-\frac{1}{3{{x}^{2/3}}(1+{{x}^{2/3}})}\] done
clear
D)
\[-\frac{a}{3{{x}^{2/3}}(1+{{x}^{2/3}})}\] done
clear
View Solution play_arrow
-
question_answer70)
If \[y={{\cot }^{-1}}\left( \frac{1+x}{1-x} \right)\], then \[\frac{dy}{dx}=\] [DSSE 1984]
A)
\[\frac{1}{1+{{x}^{2}}}\] done
clear
B)
\[-\frac{1}{1+{{x}^{2}}}\] done
clear
C)
\[\frac{2}{1+{{x}^{2}}}\] done
clear
D)
\[-\frac{2}{1+{{x}^{2}}}\] done
clear
View Solution play_arrow
-
question_answer71)
The differential coefficient of the given function \[{{\log }_{e}}\left( \sqrt{\frac{1+\sin x}{1-\sin x}} \right)\] with respect to x is [MP PET 1993]
A)
\[\cos \text{ec}\,x\] done
clear
B)
\[\tan x\] done
clear
C)
\[\cos x\] done
clear
D)
\[\sec x\] done
clear
View Solution play_arrow
-
question_answer72)
\[\frac{d}{dx}\left[ \log \sqrt{\frac{1-\cos x}{1+\cos x}} \right]=\] [BIT Ranchi 1990]
A)
\[\sec x\] done
clear
B)
\[\cos \text{ec}\,x\] done
clear
C)
\[\cos \text{ec}\frac{x}{2}\] done
clear
D)
\[\sec \frac{x}{2}\] done
clear
View Solution play_arrow
-
question_answer73)
\[\frac{d}{dx}\left[ {{\tan }^{-1}}\sqrt{\frac{1-\cos x}{1+\cos x}} \right]=\] [BIT Ranchi 1989; Roorkee 1989; RPET 1996]
A)
\[-\frac{1}{2}\] done
clear
B)
0 done
clear
C)
\[\frac{1}{2}\] done
clear
D)
1 done
clear
View Solution play_arrow
-
question_answer74)
If \[f(x)={{\tan }^{-1}}\left( \frac{\sin x}{1+\cos x} \right)\],then \[f'\left( \frac{\pi }{3} \right)=\] [BIT Ranchi 1990]
A)
\[\frac{1}{2(1+\cos x)}\] done
clear
B)
\[\frac{1}{2}\] done
clear
C)
\[\frac{1}{4}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer75)
\[\frac{d}{dx}{{e}^{x\sin x}}=\] [DSSE 1979]
A)
\[{{e}^{x\sin x}}(x\cos x+\sin x)\] done
clear
B)
\[{{e}^{x\sin x}}(\cos x+x\sin x)\] done
clear
C)
\[{{e}^{x\sin x}}(\cos x+\sin x)\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer76)
\[\frac{d}{dx}\{\log (\sec x+\tan x)\}=\] [AISSE 1982]
A)
\[\cos x\] done
clear
B)
\[\sec x\] done
clear
C)
\[\tan x\] done
clear
D)
\[\cot x\] done
clear
View Solution play_arrow
-
question_answer77)
\[\frac{d}{dx}(x{{e}^{{{x}^{2}}}})=\] [DSSE 1981]
A)
\[2{{x}^{2}}{{e}^{x}}^{2}+{{e}^{x}}^{2}\] done
clear
B)
\[{{x}^{2}}{{e}^{x}}^{2}+{{e}^{x}}^{2}\] done
clear
C)
\[{{e}^{x}}.2{{x}^{2}}+{{e}^{x}}^{2}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer78)
\[\frac{d}{dx}\left[ \frac{{{e}^{ax}}}{\sin (bx+c)} \right]=\] [AI CBSE 1983]
A)
\[\frac{{{e}^{ax}}[a\sin (bx+c)+b\cos (bx+c)]}{{{\sin }^{2}}(bx+c)}\] done
clear
B)
\[\frac{{{e}^{ax}}[a\sin (bx+c)-b\cos (bx+c)]}{\sin (bx+c)}\] done
clear
C)
\[\frac{{{e}^{ax}}[a\sin (bx+c)-b\cos (bx+c)]}{{{\sin }^{2}}(bx+c)}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer79)
If \[y=\frac{{{e}^{x}}\log x}{{{x}^{2}}}\], then \[\frac{dy}{dx}=\] [AI CBSE 1982]
A)
\[\frac{{{e}^{x}}[1+(x+2)\log x]}{{{x}^{3}}}\] done
clear
B)
\[\frac{{{e}^{x}}[1-(x-2)\log x]}{{{x}^{4}}}\] done
clear
C)
\[\frac{{{e}^{x}}[1-(x-2)\log x]}{{{x}^{3}}}\] done
clear
D)
\[\frac{{{e}^{x}}[1+(x-2)\log x]}{{{x}^{3}}}\] done
clear
View Solution play_arrow
-
question_answer80)
If \[y=\frac{{{e}^{2x}}\cos x}{x\sin x},\]then \[\frac{dy}{dx}=\] [AI CBSE 1982]
A)
\[\frac{{{e}^{2x}}[(2x-1)\cot x-x\,\text{cose}{{\text{c}}^{2}}x]}{{{x}^{2}}}\] done
clear
B)
\[\frac{{{e}^{2x}}[(2x+1)\cot x-x\,\text{cose}{{\text{c}}^{2}}x]}{{{x}^{2}}}\] done
clear
C)
\[\frac{{{e}^{2x}}[(2x-1)\cot x+x\,\text{cose}{{\text{c}}^{2}}x]}{{{x}^{2}}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer81)
\[\frac{d}{dx}\{{{e}^{-a{{x}^{2}}}}\log (\sin x)\}=\] [AI CBSE 1984]
A)
\[{{e}^{-a{{x}^{2}}}}(\cot x+2ax\log \sin x)\] done
clear
B)
\[{{e}^{-a{{x}^{2}}}}(\cot x+ax\log \sin x)\] done
clear
C)
\[{{e}^{-a{{x}^{2}}}}(\cot x-2ax\log \sin x)\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer82)
If \[y=\log x.{{e}^{(\tan x+{{x}^{2}})}},\]then \[\frac{dy}{dx}=\] [AI CBSE 1985]
A)
\[{{e}^{(\tan x+{{x}^{2}})}}\left[ \frac{1}{x}+({{\sec }^{2}}x+x)\log x \right]\] done
clear
B)
\[{{e}^{(\tan x+{{x}^{2}})}}\left[ \frac{1}{x}+({{\sec }^{2}}x-x)\log x \right]\] done
clear
C)
\[{{e}^{(\tan x+{{x}^{2}})}}\left[ \frac{1}{x}+({{\sec }^{2}}x+2x)\log x \right]\] done
clear
D)
\[{{e}^{(\tan x+{{x}^{2}})}}\left[ \frac{1}{x}+({{\sec }^{2}}x-2x)\log x \right]\] done
clear
View Solution play_arrow
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question_answer83)
If \[y=\sqrt{\frac{1+{{e}^{x}}}{1-{{e}^{x}}}}\], then \[\frac{dy}{dx}=\] [AI CBSE 1986]
A)
\[\frac{{{e}^{x}}}{(1-{{e}^{x}})\sqrt{1-{{e}^{2x}}}}\] done
clear
B)
\[\frac{{{e}^{x}}}{(1-{{e}^{x}})\sqrt{1-{{e}^{x}}}}\] done
clear
C)
\[\frac{{{e}^{x}}}{(1-{{e}^{x}})\sqrt{1+{{e}^{2x}}}}\] done
clear
D)
\[\frac{{{e}^{x}}}{(1-{{e}^{x}})\sqrt{1-{{e}^{2x}}}}\] done
clear
View Solution play_arrow
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question_answer84)
\[\frac{d}{dx}\left\{ {{e}^{x}}\log (1+{{x}^{2}}) \right\}=\] [AI CBSE 1987]
A)
\[{{e}^{x}}\left[ \log (1+{{x}^{2}})+\frac{2x}{1+{{x}^{2}}} \right]\] done
clear
B)
\[{{e}^{x}}\left[ \log (1+{{x}^{2}})-\frac{2x}{1+{{x}^{2}}} \right]\] done
clear
C)
\[{{e}^{x}}\left[ \log (1+{{x}^{2}})+\frac{x}{1+{{x}^{2}}} \right]\] done
clear
D)
\[{{e}^{x}}\left[ \log (1+{{x}^{2}})-\frac{x}{1+{{x}^{2}}} \right]\] done
clear
View Solution play_arrow
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question_answer85)
If \[y=\frac{{{e}^{2x}}+{{e}^{-2x}}}{{{e}^{2x}}-{{e}^{-2x}}}\], then \[\frac{dy}{dx}=\] [AI CBSE 1988]
A)
\[\frac{-8}{{{({{e}^{2x}}-{{e}^{-2x}})}^{2}}}\] done
clear
B)
\[\frac{8}{{{({{e}^{2x}}-{{e}^{-2x}})}^{2}}}\] done
clear
C)
\[\frac{-4}{{{({{e}^{2x}}-{{e}^{-2x}})}^{2}}}\] done
clear
D)
\[\frac{4}{{{({{e}^{2x}}-{{e}^{-2x}})}^{2}}}\] done
clear
View Solution play_arrow
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question_answer86)
If \[y=\frac{2{{(x-\sin x)}^{3/2}}}{\sqrt{x}}\], then \[\frac{dy}{dx}=\] [Roorkee 1971]
A)
\[\frac{2{{(x-\sin x)}^{3/2}}}{\sqrt{x}}\left[ \frac{3}{2}.\frac{1-\cos x}{1-\sin x}-\frac{1}{2x} \right]\] done
clear
B)
\[\frac{2{{(x-\sin x)}^{3/2}}}{\sqrt{x}}\left[ \frac{3}{2}.\frac{1-\cos x}{x-\sin x}-\frac{1}{2x} \right]\] done
clear
C)
\[\frac{2{{(x-\sin x)}^{1/2}}}{\sqrt{x}}\left[ \frac{3}{2}.\frac{1-\cos x}{x-\sin x}-\frac{1}{2x} \right]\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer87)
\[\frac{d}{dx}\left( {{\cos }^{-1}}\sqrt{\frac{1+\cos x}{2}} \right)=\] [AI CBSE 1982]
A)
1 done
clear
B)
\[\frac{1}{2}\] done
clear
C)
\[\frac{1}{3}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer88)
If \[y={{\tan }^{-1}}\left( \frac{\sqrt{a}-\sqrt{x}}{1+\sqrt{ax}} \right)\], then \[\frac{dy}{dx}=\] [AI CBSE 1988]
A)
\[\frac{1}{2(1+x)\sqrt{x}}\] done
clear
B)
\[\frac{1}{(1+x)\sqrt{x}}\] done
clear
C)
\[-\frac{1}{2(1+x)\sqrt{x}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer89)
If \[y={{\sec }^{-1}}\left( \frac{x+1}{x-1} \right)+{{\sin }^{-1}}\left( \frac{x-1}{x+1} \right)\], then \[\frac{dy}{dx}=\] [MNR 1984]
A)
0 done
clear
B)
1 done
clear
C)
2 done
clear
D)
3 done
clear
View Solution play_arrow
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question_answer90)
\[\frac{d}{dx}({{\log }_{e}}x)({{\log }_{a}}x)]=\]
A)
\[\frac{{{\log }_{a}}x}{x}\] done
clear
B)
\[\frac{{{\log }_{x}}x}{x}\] done
clear
C)
\[\frac{2\log x}{x}\] done
clear
D)
\[\frac{2{{\log }_{a}}x}{x}\] done
clear
View Solution play_arrow
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question_answer91)
\[\frac{d}{dx}\left\{ \log \left( \frac{{{e}^{x}}}{1+{{e}^{x}}} \right) \right\}=\]
A)
\[\frac{1}{1-{{e}^{x}}}\] done
clear
B)
\[-\frac{1}{1+{{e}^{x}}}\] done
clear
C)
\[-\frac{1}{1-{{e}^{x}}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer92)
\[\frac{d}{dx}\left[ \frac{2}{\pi }\sin {{x}^{0}} \right]=\]
A)
\[\frac{\pi }{180}\cos {{x}^{0}}\] done
clear
B)
\[\frac{1}{90}\cos {{x}^{0}}\] done
clear
C)
\[\frac{\pi }{90}\cos {{x}^{0}}\] done
clear
D)
\[\frac{2}{90}\cos {{x}^{0}}\] done
clear
View Solution play_arrow
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question_answer93)
\[\frac{d}{dx}\left[ \log \sqrt{\sin \sqrt{{{e}^{x}}}} \right]\]=
A)
\[\frac{1}{4}{{e}^{x/2}}\cot ({{e}^{x/2}})\] done
clear
B)
\[{{e}^{x/2}}\cot ({{e}^{x/2}})\] done
clear
C)
\[\frac{1}{4}{{e}^{x}}\cot \,({{e}^{x}})\] done
clear
D)
\[\frac{1}{2}{{e}^{x/2}}\cot \,({{e}^{x/2}})\] done
clear
View Solution play_arrow
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question_answer94)
If \[f(x)=\,|x|,\]then \[f'(0)=\] [MNR 1982]
A)
0 done
clear
B)
1 done
clear
C)
x done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer95)
At \[x=\sqrt{\frac{\pi }{2}},\frac{d}{dx}\cos (\sin {{x}^{2}})\]=
A)
?1 done
clear
B)
1 done
clear
C)
0 done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer96)
\[\frac{d}{dx}[{{\tan }^{-1}}(\cot x)+{{\cot }^{-1}}(\tan x)]=\]
A)
0 done
clear
B)
1 done
clear
C)
? 1 done
clear
D)
? 2 done
clear
View Solution play_arrow
-
question_answer97)
\[\frac{d}{dx}[{{e}^{ax}}\cos (bx+c)]\]= [AISSE 1989]
A)
\[{{e}^{ax}}[a\cos (bx+c)-b\sin (bx+c)]\] done
clear
B)
\[{{e}^{ax}}[a\sin (bx+c)-b\cos (bx+c)]\] done
clear
C)
\[{{e}^{ax}}[\cos (bx+c)-\sin (bx+c)]\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer98)
If \[y=\log \log x\], then \[{{e}^{y}}\frac{dy}{dx}=\] [MP PET 1994, 95]
A)
\[\frac{1}{x\log x}\] done
clear
B)
\[\frac{1}{x}\] done
clear
C)
\[\frac{1}{\log x}\] done
clear
D)
\[{{e}^{y}}\] done
clear
View Solution play_arrow
-
question_answer99)
If \[y={{\sin }^{-1}}\left( \frac{19}{20}x \right)+{{\cos }^{-1}}\left( \frac{19}{20}x \right)\], then \[\frac{dy}{dx}=\]
A)
0 done
clear
B)
1 done
clear
C)
? 1 done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer100)
If \[y=(1+{{x}^{1/4}})(1+{{x}^{1/2}})(1-{{x}^{1/4}})\], then \[\frac{dy}{dx}\]= [MP PET 1994]
A)
1 done
clear
B)
? 1 done
clear
C)
x done
clear
D)
\[\sqrt{x}\] done
clear
View Solution play_arrow
-
question_answer101)
If \[y=\frac{{{a}^{{{\cos }^{-1}}x}}}{1+{{a}^{{{\cos }^{-1}}x}}}\]and \[z={{a}^{{{\cos }^{-1}}x}}\], then \[\frac{dy}{dx}\]= [MP PET 1994]
A)
\[\frac{1}{1+{{a}^{{{\cos }^{-1}}x}}}\] done
clear
B)
\[-\frac{1}{1+{{a}^{{{\cos }^{-1}}x}}}\] done
clear
C)
\[\frac{1}{{{(1+{{a}^{{{\cos }^{-1}}x}})}^{2}}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer102)
If \[f(x)=(x-{{x}_{0}})g(x)\], where \[g(x)\] is continuous at \[{{x}_{0}}\], then \[f'({{x}_{0}})\] is equal to
A)
0 done
clear
B)
\[{{x}_{0}}\] done
clear
C)
\[g({{x}_{0}})\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer103)
If\[y={{\log }_{\sin x}}(\tan x),\] then \[{{\left( \frac{dy}{dx} \right)}_{\pi /4}}=\]
A)
\[\frac{4}{\log 2}\] done
clear
B)
\[-4\log 2\] done
clear
C)
\[\frac{-4}{\log 2}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer104)
If \[y={{\log }_{2}}[{{\log }_{2}}(x)]\], then \[\frac{dy}{dx}\]is equal to
A)
\[\frac{{{\log }_{2}}e}{x{{\log }_{e}}x}\] done
clear
B)
\[\frac{1}{{{\log }_{e}}x{{\log }_{e}}2}\] done
clear
C)
\[\frac{1}{{{\log }_{e}}{{(2x)}^{x}}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer105)
\[\frac{d}{dx}({{e}^{{{x}^{3}}}})\] is equal to [RPET 1995]
A)
\[3x{{e}^{{{x}^{3}}}}\] done
clear
B)
\[3{{x}^{2}}{{e}^{{{x}^{3}}}}\] done
clear
C)
\[3x{{\left( {{e}^{{{x}^{3}}}} \right)}^{2}}\] done
clear
D)
\[2{{x}^{2}}{{e}^{{{x}^{3}}}}\] done
clear
View Solution play_arrow
-
question_answer106)
\[\frac{d}{dx}({{\sin }^{-1}}x)\] is equal to [RPET 1995]
A)
\[\frac{1}{\sqrt{1-{{x}^{2}}}}\] done
clear
B)
\[-\frac{1}{\sqrt{1-{{x}^{2}}}}\] done
clear
C)
\[\frac{1}{\sqrt{1+{{x}^{2}}}}\] done
clear
D)
\[\frac{-1}{\sqrt{1+{{x}^{2}}}}\] done
clear
View Solution play_arrow
-
question_answer107)
If \[y={{\tan }^{-1}}\sqrt{\frac{1+\cos x}{1-\cos x}}\], then \[\frac{dy}{dx}\] is equal to [Roorkee 1995]
A)
0 done
clear
B)
\[-\frac{1}{2}\] done
clear
C)
½ done
clear
D)
1 done
clear
View Solution play_arrow
-
question_answer108)
If \[y=\frac{{{\sin }^{-1}}x}{\sqrt{1-{{x}^{2}}}}\], then \[(1-{{x}^{2}})\frac{dy}{dx}\] is equal to [RPET 1995]
A)
\[x+y\] done
clear
B)
\[1+xy\] done
clear
C)
1? xy done
clear
D)
\[xy-2\] done
clear
View Solution play_arrow
-
question_answer109)
Differential coefficient of \[{{\sec }^{-1}}x\] is [RPET 1995]
A)
\[\frac{1}{x\sqrt{1-{{x}^{2}}}}\] done
clear
B)
\[-\frac{1}{x\sqrt{1-{{x}^{2}}}}\] done
clear
C)
\[\frac{1}{x\sqrt{{{x}^{2}}-1}}\] done
clear
D)
\[\frac{-1}{x\sqrt{{{x}^{2}}-1}}\] done
clear
View Solution play_arrow
-
question_answer110)
If \[f(2)=4\], \[f'(2)=1\]then \[\underset{x\to 2}{\mathop{\lim }}\,\frac{xf(2)-2f(x)}{x-2}=\] [RPET 1995, 2000]
A)
1 done
clear
B)
2 done
clear
C)
3 done
clear
D)
?2 done
clear
View Solution play_arrow
-
question_answer111)
\[\frac{d}{dx}\left[ \log \left( x+\frac{1}{x} \right) \right]=\] [MP PET 1995]
A)
\[\left( x+\frac{1}{x} \right)\] done
clear
B)
\[\frac{\left( 1+\frac{1}{{{x}^{2}}} \right)}{\left( 1+\frac{1}{x} \right)}\] done
clear
C)
\[\frac{\left( 1-\frac{1}{{{x}^{2}}} \right)}{\left( x+\frac{1}{x} \right)}\] done
clear
D)
\[\left( 1+\frac{1}{x} \right)\] done
clear
View Solution play_arrow
-
question_answer112)
If \[y={{\sin }^{-1}}\sqrt{x}\], then \[\frac{dy}{dx}=\] [MP PET 1995]
A)
\[\frac{2}{\sqrt{x}\sqrt{1-x}}\] done
clear
B)
\[\frac{-2}{\sqrt{x}\sqrt{1-x}}\] done
clear
C)
\[\frac{1}{2\sqrt{x}\sqrt{1-x}}\] done
clear
D)
\[\frac{1}{\sqrt{1-x}}\] done
clear
View Solution play_arrow
-
question_answer113)
If \[y={{\sin }^{-1}}\sqrt{(1-x)}+{{\cos }^{-1}}\sqrt{x}\], then \[\frac{dy}{dx}=\]
A)
\[\frac{1}{\sqrt{x(1-x)}}\] done
clear
B)
\[\frac{-1}{\sqrt{x(1-x)}}\] done
clear
C)
\[\frac{1}{\sqrt{x(1+x)}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer114)
If \[y={{x}^{n}}\log x+x{{(\log x)}^{n}}\], then \[\frac{dy}{dx}=\]
A)
\[{{x}^{n-1}}(1+n\log x)+{{(\log x)}^{n-1}}[n+\log x]\] done
clear
B)
\[{{x}^{n-2}}(1+n\log x)+{{(\log x)}^{n-1}}[n+\log x]\] done
clear
C)
\[{{x}^{n-1}}(1+n\log x)+{{(\log x)}^{n-1}}[n-\log x]\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer115)
If \[y\sqrt{{{x}^{2}}+1}=\log \left\{ \sqrt{{{x}^{2}}+1}-x \right\}\], then\[({{x}^{2}}+1)\frac{dy}{dx}+xy+1=\] [Roorkee 1978; Kurukshetra CEE 1998]
A)
0 done
clear
B)
1 done
clear
C)
2 done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer116)
The derivative of tanx ? x with respect to x is [SCRA 1996]
A)
\[1-{{\tan }^{2}}x\] done
clear
B)
tan x done
clear
C)
\[-{{\tan }^{2}}x\] done
clear
D)
\[{{\tan }^{2}}x\] done
clear
View Solution play_arrow
-
question_answer117)
If \[f(x)=({{\log }_{\cot x}}\tan x){{({{\log }_{\tan x}}\cot x)}^{-1}},\]then \[f'(2)=\]
A)
2 done
clear
B)
0 done
clear
C)
\[\frac{1}{2}\] done
clear
D)
? 2 done
clear
View Solution play_arrow
-
question_answer118)
If \[f(x)=3{{e}^{{{x}^{2}}}}\],then \[f'(x)-2xf(x)+\frac{1}{3}f(0)-f'(0)=\]
A)
0 done
clear
B)
1 done
clear
C)
\[\frac{7}{3}{{e}^{{{x}^{2}}}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer119)
If \[y={{\log }_{\cos x}}\sin x\], then \[\frac{dy}{dx}\]is equal to
A)
\[\frac{\cot x\log \cos x+\tan x\log \sin x}{{{(\log \cos x)}^{2}}}\] done
clear
B)
\[\frac{\tan x\log \cos x+\cot x\log \sin x}{{{(\log \cos x)}^{2}}}\] done
clear
C)
\[\frac{\cot x\log \cos x+\tan x\log \sin x}{{{(\log \sin x)}^{2}}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer120)
\[\frac{d}{dx}\left[ \log \left\{ {{e}^{x}}{{\left( \frac{x+2}{x-2} \right)}^{3/4}} \right\} \right]\] equals
A)
\[\frac{{{x}^{2}}-7}{{{x}^{2}}-4}\] done
clear
B)
1 done
clear
C)
\[\frac{{{x}^{2}}+1}{{{x}^{2}}-4}\] done
clear
D)
\[{{e}^{x}}\frac{{{x}^{2}}-1}{{{x}^{2}}-4}\] done
clear
View Solution play_arrow
-
question_answer121)
For the curve \[\sqrt{x}+\sqrt{y}=1,\frac{dy}{dx}\]at \[\left( \frac{1}{4},\frac{1}{4} \right)\]is [Karnataka CET 1993]
A)
½ done
clear
B)
1 done
clear
C)
?1 done
clear
D)
2 done
clear
View Solution play_arrow
-
question_answer122)
Differential coefficient of \[\sqrt{\sec \sqrt{x}}\]is [MP PET 1996]
A)
\[\frac{1}{4\sqrt{x}}{{(\sec \sqrt{x})}^{3/2}}\sin \sqrt{x}\] done
clear
B)
\[\frac{1}{4\sqrt{x}}\sec \sqrt{x}\sin \sqrt{x}\] done
clear
C)
\[\frac{1}{2}\sqrt{x}{{(\sec \sqrt{x})}^{3/2}}\sin \sqrt{x}\] done
clear
D)
\[\frac{1}{2}\sqrt{x}\sec \sqrt{x}\sin \sqrt{x}\] done
clear
View Solution play_arrow
-
question_answer123)
If \[y={{e}^{(1+{{\log }_{e}}x)}}\], then the value of \[\frac{dy}{dx}=\] [MP PET 1996; Pb. CET 2001]
A)
e done
clear
B)
1 done
clear
C)
0 done
clear
D)
\[{{\log }_{e}}x\,\,{{e}^{{{\log }_{e}}ex}}\] done
clear
View Solution play_arrow
-
question_answer124)
For the function \[f(x)={{x}^{2}}-6x+8,2\le x\le 4\], the value of x for which \[f'(x)\] vanishes, is [MP PET 1996]
A)
\[\frac{9}{4}\] done
clear
B)
\[\frac{5}{2}\] done
clear
C)
3 done
clear
D)
\[\frac{7}{2}\] done
clear
View Solution play_arrow
-
question_answer125)
If \[f(x)={{e}^{x}}g(x),g(0)=2,g'(0)=1\], then \[f'(0)\]is
A)
1 done
clear
B)
3 done
clear
C)
2 done
clear
D)
0 done
clear
View Solution play_arrow
-
question_answer126)
If \[y={{e}^{x}}\log x\], then \[\frac{dy}{dx}\]is [SCRA 1996]
A)
\[\frac{{{e}^{x}}}{x}\] done
clear
B)
\[{{e}^{x}}\left( \frac{1}{x}+x\log x \right)\] done
clear
C)
\[{{e}^{x}}\left( \frac{1}{x}+\log x \right)\] done
clear
D)
\[\frac{{{e}^{x}}}{\log x}\] done
clear
View Solution play_arrow
-
question_answer127)
If \[y={{\cot }^{-1}}\left[ \frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}} \right]\], then \[\frac{dy}{dx}=\]
A)
\[\frac{1}{2}\] done
clear
B)
\[\frac{2}{3}\] done
clear
C)
3 done
clear
D)
1 done
clear
View Solution play_arrow
-
question_answer128)
If \[y=\sec {{x}^{0}}\], then \[\frac{dy}{dx}=\] [MP PET 1997]
A)
\[\sec x\tan x\] done
clear
B)
\[\sec {{x}^{o}}\tan {{x}^{o}}\] done
clear
C)
\[\frac{\pi }{180}\sec {{x}^{o}}\tan {{x}^{o}}\] done
clear
D)
\[\frac{180}{\pi }\sec {{x}^{o}}\tan {{x}^{o}}\] done
clear
View Solution play_arrow
-
question_answer129)
If \[y=\sqrt{\sin \sqrt{x}}\], then \[\frac{dy}{dx}=\] [MP PET 1997]
A)
\[\frac{1}{2\sqrt{\cos \sqrt{x}}}\] done
clear
B)
\[\frac{\sqrt{\cos \sqrt{x}}}{2x}\] done
clear
C)
\[\frac{\cos \sqrt{x}}{4\sqrt{x}\sqrt{\sin \sqrt{x}}}\] done
clear
D)
\[\frac{1}{2\sqrt{\sin x}}\] done
clear
View Solution play_arrow
-
question_answer130)
If\[y={{\log }_{10}}{{x}^{2}}\], then \[\frac{dy}{dx}\] is equal to
A)
\[\frac{2}{x}\] done
clear
B)
\[\frac{2}{x{{\log }_{e}}10}\] done
clear
C)
\[\frac{1}{x{{\log }_{e}}10}\] done
clear
D)
\[\frac{1}{10x}\] done
clear
View Solution play_arrow
-
question_answer131)
If \[y={{3}^{{{x}^{2}}}}\], then \[\frac{dy}{dx}\] is equal to
A)
\[({{x}^{2}}){{3}^{{{x}^{2}}-1}}\] done
clear
B)
\[3{{x}^{2}}.2x\] done
clear
C)
\[{{3}^{{{x}^{2}}}}.2x.\log 3\] done
clear
D)
\[({{x}^{2}}-1).3\] done
clear
View Solution play_arrow
-
question_answer132)
The first derivative of the function \[(\sin 2x\cos 2x\cos 3x+{{\log }_{2}}{{2}^{x+3}})\] with respect to x at \[x=\pi \]is [MP PET 1998]
A)
2 done
clear
B)
?1 done
clear
C)
\[-2+{{2}^{\pi }}{{\log }_{e}}2\] done
clear
D)
\[-2+{{\log }_{e}}2\] done
clear
View Solution play_arrow
-
question_answer133)
The values of x, at which the first derivative of the function \[{{\left( \sqrt{x}+\frac{1}{\sqrt{x}} \right)}^{2}}\]w.r.t. x is \[\frac{3}{4}\], are [MP PET 1998]
A)
\[\pm \,2\] done
clear
B)
\[\pm \frac{1}{2}\] done
clear
C)
\[\pm \frac{\sqrt{3}}{2}\] done
clear
D)
\[\pm \frac{2}{\sqrt{3}}\] done
clear
View Solution play_arrow
-
question_answer134)
If \[y=\frac{{{(1-x)}^{2}}}{{{x}^{2}}}\], then \[\frac{dy}{dx}\]is [MP PET 1999]
A)
\[\frac{2}{{{x}^{2}}}+\frac{2}{{{x}^{3}}}\] done
clear
B)
\[-\frac{2}{{{x}^{2}}}+\frac{2}{{{x}^{3}}}\] done
clear
C)
\[-\frac{2}{{{x}^{2}}}-\frac{2}{{{x}^{3}}}\] done
clear
D)
\[-\frac{2}{{{x}^{3}}}+\frac{2}{{{x}^{2}}}\] done
clear
View Solution play_arrow
-
question_answer135)
If \[pv=81\], then \[\frac{dp}{dv}\] is at v = 9 equal to [MP PET 1999]
A)
1 done
clear
B)
?1 done
clear
C)
2 done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer136)
\[\frac{d}{dx}(\sin 2{{x}^{2}})\] equals [RPET 1996]
A)
\[4x\cos \,(2{{x}^{2}})\] done
clear
B)
\[2\sin {{x}^{2}}\cos {{x}^{2}}\] done
clear
C)
\[4x\sin ({{x}^{2}})\] done
clear
D)
\[4x\sin ({{x}^{2}})\cos ({{x}^{2}})\] done
clear
View Solution play_arrow
-
question_answer137)
\[\frac{d}{dx}\cos \,{{\text{h}}^{-1}}(\sec x)=\] [RPET 1997]
A)
sec x done
clear
B)
sin x done
clear
C)
tan x done
clear
D)
cosec x done
clear
View Solution play_arrow
-
question_answer138)
If \[f(x)=\frac{1}{\sqrt{{{x}^{2}}+{{a}^{2}}}+\sqrt{{{x}^{2}}+{{b}^{2}}}}\], then \[{f}'(x)\] is equal to [Kurukshetra CEE 1998]
A)
\[\frac{x}{({{a}^{2}}-{{b}^{2}})}\left[ \frac{1}{\sqrt{{{x}^{2}}+{{a}^{2}}}}-\frac{1}{\sqrt{{{x}^{2}}+{{b}^{2}}}} \right]\] done
clear
B)
\[\frac{x}{({{a}^{2}}+{{b}^{2}})}\left[ \frac{1}{\sqrt{{{x}^{2}}+{{a}^{2}}}}-\frac{2}{\sqrt{{{x}^{2}}+{{b}^{2}}}} \right]\] done
clear
C)
\[\frac{x}{({{a}^{2}}-{{b}^{2}})}\left[ \frac{1}{\sqrt{{{x}^{2}}+{{a}^{2}}}}+\frac{1}{\sqrt{{{x}^{2}}+{{b}^{2}}}} \right]\] done
clear
D)
\[({{a}^{2}}+{{b}^{2}})\left[ \frac{1}{\sqrt{{{x}^{2}}+{{a}^{2}}}}-\frac{2}{\sqrt{{{x}^{2}}+{{b}^{2}}}} \right]\] done
clear
View Solution play_arrow
-
question_answer139)
The derivative of \[f(x)=x|x|\] is [SCRA 1996]
A)
\[2x\] done
clear
B)
? 2x done
clear
C)
\[2{{x}^{2}}\] done
clear
D)
\[2|x|\] done
clear
View Solution play_arrow
-
question_answer140)
The derivative of \[y=1-|x|\]at \[x=0\]is [SCRA 1996]
A)
0 done
clear
B)
1 done
clear
C)
?1 done
clear
D)
Does not exist done
clear
View Solution play_arrow
-
question_answer141)
The derivative of \[\sqrt{\sqrt{x}+1}\]is [SCRA 1996]
A)
\[\frac{1}{\sqrt{x}(\sqrt{x}+1)}\] done
clear
B)
\[\frac{1}{\sqrt{x}\sqrt{x+1}}\] done
clear
C)
\[\frac{4}{\sqrt{x(\sqrt{x}+1)}}\] done
clear
D)
\[\frac{1}{4\sqrt{x(\sqrt{x}+1)}}\] done
clear
View Solution play_arrow
-
question_answer142)
If \[y={{e}^{\sqrt{x}}}\], then \[\frac{dy}{dx}\]equals [SCRA 1996]
A)
\[\frac{{{e}^{\sqrt{x}}}}{2\sqrt{x}}\] done
clear
B)
\[\frac{\sqrt{x}}{{{e}^{\sqrt{x}}}}\] done
clear
C)
\[\frac{x}{{{e}^{\sqrt{x}}}}\] done
clear
D)
\[\frac{2\sqrt{x}}{{{e}^{\sqrt{x}}}}\] done
clear
View Solution play_arrow
-
question_answer143)
If \[f(x)={{\cos }^{-1}}\left[ \frac{1-{{(\log x)}^{2}}}{1+{{(\log x)}^{2}}} \right]\,,\]then the value of \[f'(e)=\] [Karnataka CET 1999; Pb. CET 2000]
A)
1 done
clear
B)
1/e done
clear
C)
2/e done
clear
D)
\[\frac{2}{{{e}^{2}}}\] done
clear
View Solution play_arrow
-
question_answer144)
The derivative of \[f(x)=\,|{{x}^{2}}-x|\] at x = 2 is [AMU 1999]
A)
? 3 done
clear
B)
0 done
clear
C)
3 done
clear
D)
Not defined done
clear
View Solution play_arrow
-
question_answer145)
If \[f(1)=3,\,{f}'(1)=2,\]then \[\frac{d}{dx}\{\log f\,({{e}^{x}}+2x)\}\] at \[x=0\] is [AMU 1999]
A)
2 / 3 done
clear
B)
3 / 2 done
clear
C)
2 done
clear
D)
0 done
clear
View Solution play_arrow
-
question_answer146)
\[\frac{d}{dx}{{\log }_{\sqrt{x}}}(1/x)\]is equal to [AMU 1999]
A)
\[-\frac{1}{2\sqrt{x}}\] done
clear
B)
? 2 done
clear
C)
\[-\frac{1}{{{x}^{2}}\sqrt{x}}\] done
clear
D)
0 done
clear
View Solution play_arrow
-
question_answer147)
The value of \[\frac{d}{dx}[|x-1|+|x-5|]\] at \[x=3\] is [MP PET 2000]
A)
? 2 done
clear
B)
0 done
clear
C)
2 done
clear
D)
4 done
clear
View Solution play_arrow
-
question_answer148)
\[\frac{d}{dx}\left[ \left( \frac{{{\tan }^{2}}2x-{{\tan }^{2}}x}{1-{{\tan }^{2}}2x{{\tan }^{2}}x} \right)\cot 3x \right]\] [AMU 2000]
A)
\[\tan 2x\,\tan x\] done
clear
B)
\[\tan 3x\tan x\] done
clear
C)
\[{{\sec }^{2}}x\] done
clear
D)
\[\sec x\tan x\] done
clear
View Solution play_arrow
-
question_answer149)
If \[y={{\tan }^{-1}}\left( \frac{\sqrt{x}-x}{1+{{x}^{3/2}}} \right),\]then \[y'(1)\] is [AMU 2000]
A)
0 done
clear
B)
\[\frac{1}{2}\] done
clear
C)
? 1 done
clear
D)
\[-\frac{1}{4}\] done
clear
View Solution play_arrow
-
question_answer150)
\[{{10}^{-x\,\tan x}}\left[ \frac{d}{dx}({{10}^{x\tan x}}) \right]\] is equal to [AMU 2000]
A)
\[\tan x\,+x\,\,{{\sec }^{2}}x\] done
clear
B)
\[\ln \,10\,(\tan x+x{{\sec }^{2}}x)\] done
clear
C)
\[\ln \,10\,\left( \tan x+\frac{x}{{{\cos }^{2}}x}+\tan x\sec x \right)\] done
clear
D)
\[x\tan x\,\text{ln}\,\,10\] done
clear
View Solution play_arrow
-
question_answer151)
If \[f(x)\] has a derivative at \[x=a,\]then \[\underset{x\to a}{\mathop{\lim }}\,\frac{xf(a)-af(x)}{x-a}\] is equal to [AMU 2000]
A)
\[f(a)-a\,f\,'(a)\] done
clear
B)
\[a\,f(a)-f\,'(a)\] done
clear
C)
\[f(a)+f'(a)\] done
clear
D)
\[a\,f(a)+f\,'(a)\] done
clear
View Solution play_arrow
-
question_answer152)
Given that \[\frac{d}{dx}f(x)=f\,'(x)\]. The relationship \[f\,'(a+b)=f\,'(a)+f\,'(b)\] is valid if \[f(x)\] is equal to [AMU 2000]
A)
\[x\] done
clear
B)
\[{{x}^{2}}\] done
clear
C)
\[{{x}^{3}}\] done
clear
D)
\[{{x}^{4}}\] done
clear
View Solution play_arrow
-
question_answer153)
The derivative of \[f(x)=|x{{|}^{3}}\] at \[x=0\] is [RPET 2001; Kurukshetra CEE 2002]
A)
0 done
clear
B)
1 done
clear
C)
?1 done
clear
D)
Not defined done
clear
View Solution play_arrow
-
question_answer154)
If \[y=\sqrt{\sin x+y},\] then \[\frac{dy}{dx}\] equals to [RPET 2001]
A)
\[\frac{\sin x}{2y-1}\] done
clear
B)
\[\frac{\cos x}{2y-1}\] done
clear
C)
\[\frac{\sin x}{2y+1}\] done
clear
D)
\[\frac{\cos x}{2y+1}\] done
clear
View Solution play_arrow
-
question_answer155)
If \[y=(1+{{x}^{2}}){{\tan }^{-1}}x-x,\]then \[\frac{dy}{dx}=\] [Karnataka CET 2001]
A)
\[{{\tan }^{-1}}x\] done
clear
B)
\[2x{{\tan }^{-1}}x\] done
clear
C)
\[2x{{\tan }^{-1}}x-1\] done
clear
D)
\[\frac{2x}{{{\tan }^{-1}}x}\] done
clear
View Solution play_arrow
-
question_answer156)
If \[x=y\sqrt{1-{{y}^{2}},}\]then \[\frac{dy}{dx}=\] [MP PET 2001]
A)
0 done
clear
B)
x done
clear
C)
\[\frac{\sqrt{1-{{y}^{2}}}}{1-2{{y}^{2}}}\] done
clear
D)
\[\frac{\sqrt{1-{{y}^{2}}}}{1+2{{y}^{2}}}\] done
clear
View Solution play_arrow
-
question_answer157)
If \[y={{\tan }^{-1}}\left[ \frac{\sin x+\cos x}{\cos x-\sin x} \right]\,,\] then \[\frac{dy}{dx}\] is [UPSEAT 2001]
A)
\[1/2\] done
clear
B)
\[\pi /4\] done
clear
C)
0 done
clear
D)
1 done
clear
View Solution play_arrow
-
question_answer158)
If \[y=\frac{a+b{{x}^{3/2}}}{{{x}^{5/4}}}\] and \[{y}'=0\] at \[x=5\], then the ratio \[a:b\] is equal to [AMU 2001]
A)
\[\sqrt{5}:1\] done
clear
B)
5 : 2 done
clear
C)
3 : 5 done
clear
D)
1 : 2 done
clear
View Solution play_arrow
-
question_answer159)
\[\frac{d}{dx}\left[ {{\tan }^{-1}}\left( \frac{a-x}{1+ax} \right) \right]=\] [Karnataka CET 2001; Pb. CET 2001]
A)
\[-\frac{1}{1+{{x}^{2}}}\] done
clear
B)
\[\frac{1}{1+{{a}^{2}}}-\frac{1}{1+{{x}^{2}}}\] done
clear
C)
\[\frac{1}{1+{{\left( \frac{a-x}{1+ax} \right)}^{2}}}\] done
clear
D)
\[\frac{-1}{\sqrt{1-{{\left( \frac{a-x}{1+ax} \right)}^{2}}}}\] done
clear
View Solution play_arrow
-
question_answer160)
\[\frac{d}{dx}\left[ \log \left\{ {{e}^{x}}{{\left( \frac{x-2}{x+2} \right)}^{3/4}} \right\} \right]\] equals to [RPET 2001]
A)
1 done
clear
B)
\[\frac{{{x}^{2}}+1}{{{x}^{2}}-4}\] done
clear
C)
\[\frac{{{x}^{2}}-1}{{{x}^{2}}-4}\] done
clear
D)
\[{{e}^{x}}\frac{{{x}^{2}}-1}{{{x}^{2}}-4}\] done
clear
View Solution play_arrow
-
question_answer161)
If \[y=\sec ({{\tan }^{-1}}x),\]then \[\frac{dy}{dx}\] is [DCE 2002; Kurukshetra CEE 2001]
A)
\[\frac{x}{\sqrt{1+{{x}^{2}}}}\] done
clear
B)
\[\frac{-x}{\sqrt{1+{{x}^{2}}}}\] done
clear
C)
\[\frac{x}{\sqrt{1-{{x}^{2}}}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer162)
The differential coefficient of the function \[|x-1|+|x-3|\] at the point \[x=2\] is [RPET 2002; Pb. CET 2000, 04]
A)
? 2 done
clear
B)
0 done
clear
C)
2 done
clear
D)
Undefined done
clear
View Solution play_arrow
-
question_answer163)
If \[f(x)\] is a differentiable function, then \[\underset{x\to a}{\mathop{\lim }}\,\frac{af(x)-xf(a)}{x-a}\] is [UPSEAT 2002]
A)
\[a{f}'\,(a)-f\,(a)\] done
clear
B)
\[af\,(a)-f'(a)\] done
clear
C)
\[a{f}'\,(a)+f\,(a)\] done
clear
D)
\[af\,(a)+f'(a)\] done
clear
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question_answer164)
If \[x=\exp \left\{ {{\tan }^{-1}}\left( \frac{y-{{x}^{2}}}{{{x}^{2}}} \right) \right\}\,\,\], then \[\frac{dy}{dx}\]equals [MP PET 2002]
A)
\[2x\,[1+\tan \,(\log x)]+x{{\sec }^{2}}(\log x)\] done
clear
B)
\[x\,[1+\tan \,(\log x)]+{{\sec }^{2}}(\log x)\] done
clear
C)
\[2x\,[1+\tan \,(\log x)]+{{x}^{2}}\,\,{{\sec }^{2}}(\log x)\] done
clear
D)
\[2x\,[1+\tan \,(\log x)]+{{\sec }^{2}}(\log x)\] done
clear
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question_answer165)
If \[f(x)=\sqrt{ax}+\frac{{{a}^{2}}}{\sqrt{ax}},\]then \[f'(a)=\] [EAMCET 2002]
A)
? 1 done
clear
B)
1 done
clear
C)
0 done
clear
D)
a done
clear
View Solution play_arrow
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question_answer166)
Derivative of \[{{x}^{6}}+{{6}^{x}}\]with respect to x is [Kerala (Engg.) 2002]
A)
\[12x\] done
clear
B)
\[x+4\] done
clear
C)
\[6{{x}^{5}}+{{6}^{x}}\,\,\log 6\] done
clear
D)
\[6{{x}^{5}}+x{{6}^{x-1}}\] done
clear
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question_answer167)
If \[\sin y+{{e}^{-x\,\cos y}}=e,\]then \[\frac{dy}{dx}\] at \[(1,\pi )\] is [Kerala (Engg.) 2002]
A)
\[\sin y\] done
clear
B)
\[-x\cos y\] done
clear
C)
\[e\] done
clear
D)
\[\sin y-x\,\cos y\] done
clear
View Solution play_arrow
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question_answer168)
The derivative of \[f(x)=3|2+x|\] at the point \[{{x}_{0}}=-3\] is [Orissa JEE 2002]
A)
3 done
clear
B)
? 3 done
clear
C)
0 done
clear
D)
Does not exist done
clear
View Solution play_arrow
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question_answer169)
Derivative of the function \[f(x)={{\log }_{5}}({{\log }_{7}}x)\], \[x>7\] is [Orissa JEE 2002]
A)
\[\frac{1}{x(\text{In}\,\text{5)(In}\,\text{7)(lo}{{\text{g}}_{\text{7}}}x)}\] done
clear
B)
\[\frac{1}{x(\text{ln}\,\text{5)(ln}\,\text{7)}}\] done
clear
C)
\[\frac{1}{x(In\,x)}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer170)
If \[y={{\cot }^{-1}}({{x}^{2}})\], then \[\frac{dy}{dx}\] is equal to [Pb. CET 2002]
A)
\[\frac{2x}{1+{{x}^{4}}}\] done
clear
B)
\[\frac{2x}{\sqrt{1+4x}}\] done
clear
C)
\[\frac{-2x}{1+{{x}^{4}}}\] done
clear
D)
\[\frac{-2x}{\sqrt{1+{{x}^{2}}}}\] done
clear
View Solution play_arrow
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question_answer171)
If \[y=\log \tan \sqrt{x}\]then the value of \[\frac{dy}{dx}\] is [Pb. CET 2000]
A)
\[\frac{1}{2\sqrt{x}}\] done
clear
B)
\[\frac{{{\sec }^{2}}\sqrt{x}}{\sqrt{x}\tan x}\] done
clear
C)
\[2{{\sec }^{2}}\sqrt{x}\] done
clear
D)
\[\frac{{{\sec }^{2}}\sqrt{x}}{2\sqrt{x}\tan \sqrt{x}}\] done
clear
View Solution play_arrow
-
question_answer172)
If \[y={{(\cos {{x}^{2}})}^{2}}\]then \[\frac{dy}{dx}\]is equal to [Pb. CET 2004]
A)
\[-4x\sin 2{{x}^{2}}\] done
clear
B)
\[-x\sin {{x}^{2}}\] done
clear
C)
\[-2x\sin 2{{x}^{2}}\] done
clear
D)
\[-x\cos 2{{x}^{2}}\] done
clear
View Solution play_arrow
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question_answer173)
If \[y={{\tan }^{-1}}(\sec x-\tan x)\]then \[\frac{dy}{dx}=\] [Karnataka CET 2004]
A)
2 done
clear
B)
?2 done
clear
C)
½ done
clear
D)
?1/2 done
clear
View Solution play_arrow
-
question_answer174)
If \[y={{\cos }^{-1}}\cos (|x|-f(x)),\] where \[\] \[f(x)\left\{ \begin{align} & =1\,,\,\text{if}\,\,\,x>0 \\ & =-1\,,\,\text{if}\,\,\,x<0 \\ & =0\,,\,\text{if}\,\,\,x=0 \\ \end{align} \right.\], then \[{{\left. \frac{dy}{dx} \right|}_{x=\frac{5\pi }{4}}}\]is [J & K 2005]
A)
? 1 done
clear
B)
1 done
clear
C)
0 done
clear
D)
Indeterminate done
clear
View Solution play_arrow
-
question_answer175)
If \[{{x}^{m}}{{y}^{n}}={{(x+y)}^{m+n}}\]then \[{{\left. \frac{dy}{dx} \right|}_{x=1,y=2}}\] is equal to [J & K 2005]
A)
½ done
clear
B)
2 done
clear
C)
2m/n done
clear
D)
m/ 2n done
clear
View Solution play_arrow
-
question_answer176)
If \[y=\frac{{{e}^{x}}+{{e}^{-x}}}{{{e}^{x}}-{{e}^{-x}}}\] then \[\frac{dy}{dx}\] is equal to [Karnataka CET 2005]
A)
\[\sec {{\text{h}}^{2}}x\] done
clear
B)
\[\text{cosec}{{\text{h}}^{2}}x\] done
clear
C)
\[-\sec \,{{\text{h}}^{2}}x\] done
clear
D)
\[-\text{cosec}{{\text{h}}^{2}}x\] done
clear
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question_answer177)
The derivative of function \[f(x)\] is \[{{\tan }^{4}}x\]. If \[f(0)=0\] then \[\underset{x\to 0}{\mathop{\lim }}\,\frac{f(x)}{x}\]is equal to [J & K 2005]
A)
1 done
clear
B)
0 done
clear
C)
?1 done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer178)
Let \[f(x)\] be a polynomial function of the second degree. If \[f(1)=f(-1)\] and \[{{a}_{1}},{{a}_{2}},{{a}_{3}}\] are in A.P. then \[{f}'({{a}_{1}})\], \[{f}'({{a}_{2}})\], \[{f}'({{a}_{3}})\] are in [AMU 2005]
A)
A.P done
clear
B)
G.P. done
clear
C)
H.P. done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer179)
\[\frac{d}{dx}\,\,\left[ {{\tan }^{-1}}\left( \frac{\sqrt{x}(3-x)}{1-3x} \right) \right]\]= [Kerala (Engg.) 2005]
A)
\[\frac{1}{2(1+x)\,\sqrt{x}}\] done
clear
B)
\[\frac{3}{(1+x)\,\sqrt{x}}\] done
clear
C)
\[\frac{2}{(1+x)\,\sqrt{x}}\] done
clear
D)
\[\frac{2\sqrt{2}y-3}{2\sqrt{2}}=\frac{-3\sqrt{2}\times +3}{2\sqrt{2}}\] done
clear
E)
\[\frac{3}{2(1+x)\sqrt{x}}\] done
clear
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question_answer180)
If \[r={{[2\varphi +{{\cos }^{2}}(2\varphi +\pi /4)]}^{1/2}}\] then what is the value of the derivative of \[dr/d\varphi \] at \[\varphi =\pi /4\] [Orissa JEE 2005]
A)
\[2\,{{\left( \frac{1}{\pi +1} \right)}^{1/2}}\] done
clear
B)
\[2\,{{\left( \frac{2}{\pi +1} \right)}^{-1/2}}\] done
clear
C)
\[2\,{{\left( \frac{1}{\pi +1} \right)}^{-1/2}}\] done
clear
D)
\[2\,{{\left( \frac{2}{\pi +1} \right)}^{1/2}}\] done
clear
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question_answer181)
If \[f(x)=\cos x\cos 2x\cos 4x\cos 8x\cos 16x\], then \[{f}'\left( \frac{\pi }{4} \right)\] is [AMU 2005]
A)
\[\sqrt{2}\] done
clear
B)
\[\frac{1}{\sqrt{2}}\] done
clear
C)
1 done
clear
D)
\[\frac{\sqrt{3}}{2}\] done
clear
View Solution play_arrow
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question_answer182)
The derivative of \[y=(1-x)\,(2-x)....(n-x)\] at \[x=1\] is equal to [Kerala (Engg.) 2005]
A)
0 done
clear
B)
\[(-1)\,(n-1)\,!\] done
clear
C)
\[n!\,-\,1\] done
clear
D)
\[{{(-1)}^{n-1}}(n-1)\,!\] done
clear
E)
\[{{(-1)}^{n}}\,(n-1)\,!\] done
clear
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question_answer183)
If \[y={{\tan }^{-1}}\left( \frac{a\cos x-b\sin x}{b\cos x+a\sin x} \right)\] then \[\frac{dy}{dx}=\] [Kerala (Engg.) 2005]
A)
2 done
clear
B)
? 1 done
clear
C)
\[\frac{a}{b}\] done
clear
D)
0 done
clear
E)
\[\frac{b}{a}\] done
clear
View Solution play_arrow