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question_answer1) If the derivative of f\[\left( \tan x \right)\] w.r.t. \[g\left( \sec \,\,x \right)\]at \[x=\pi /4,\] where \[f'\left( 1 \right)=2\] and \[g'\left( \sqrt{2} \right)=4\] is \[\frac{\sqrt{2}}{k}\] then find k?
question_answer2) Let \[\phi \left( x \right)\] be the inverse of the function \[f\left( x \right)\] and \[{f}'\left( x \right)=\frac{1}{1+{{x}^{5}}}\,\,\text{and}\,\,\frac{d}{dx}\phi \left( x \right)={{\left[ 1+\phi \left( x \right) \right]}^{n}}\] then find n.
question_answer3) Let y be a function of x, such that \[\log \,\left( x+y \right)-2xy=0,\] then find \[y'\left( 0 \right)\].
question_answer4) Let \[f\left( x \right),g\left( x \right)\] be two continuously differentiable function satisfying the relationships \[f'\left( x \right)=g\left( x \right)\] and \[f''\left( x \right)=-f\left( x \right).\] Let \[h\left( x \right)={{\left[ f\left( x \right) \right]}^{2}}+{{\left[ g\left( x \right) \right]}^{2}}.\] If \[h\left( 0 \right)=5,\] then find value of \[h\left( 10 \right)\].
question_answer5) If \[5f\left( x \right)+3f\left( 1/x \right)=x+2,\] then find \[\frac{d}{dx}\left( x.\,f\left( x \right) \right)\] at \[x=1.\]
question_answer6) If \[y={{\tan }^{-1}}\left( \frac{{{2}^{x}}}{1+{{2}^{2x+1}}} \right),\] then \[\frac{dy}{dx}\] at \[x=0\] is \[k\,\,\log \frac{1}{2}\] then find k.
question_answer7) Find the first derivative of the function \[\left[ {{\cos }^{-1}}\left( \sin \sqrt{\frac{1+x}{2}} \right)+{{x}^{x}} \right]\] with respect to x at \[x=1\].
question_answer8) Let \[g\left( x \right)\] be inverse of function \[f\left( x \right)={{x}^{3}}+2{{x}^{2}}+4x+\sin \left( \frac{\pi }{2}x \right)\] and \[g'\left( 8 \right)\] is equal to \[\frac{1}{k}\] then find k.
question_answer9) Find \[\frac{d}{dx}\left[ {{\sin }^{2}}{{\cot }^{-1}}\left\{ \sqrt{\frac{1-x}{1+x}} \right\} \right].\]
question_answer10) If \[y={{\tan }^{-1}}\frac{\log \left( e/{{x}^{2}} \right)}{\log \left( e{{x}^{2}} \right)}+{{\tan }^{-1}}\frac{3+2\log x}{1-6\log x},\]then find\[\frac{{{d}^{2}}y}{d{{x}^{2}}}\].
question_answer11) If \[f\left( x \right)={{\cot }^{-1}}\left( \frac{{{x}^{x}}-{{x}^{-x}}}{2} \right),\]then find\[f'\left( 1 \right)\].
question_answer12) If \[y={{2}^{ax}}\]and \[\frac{dy}{dx}=\log \,\,256\]at \[x=1,\]then find value of a.
question_answer13) If \[f\left( x \right)=\cos \left( {{x}^{2}}-2\left[ x \right] \right)\]for \[0<x<1\]and \[f'\left( \frac{\sqrt{\pi }}{2} \right)=-\sqrt{\frac{\pi }{k}},\]then find k.
question_answer14) If \[y={{\log }_{{{e}^{x}}}}{{\left( x-3 \right)}^{2}}\]and \[x\ne 0,\]then find \[{{\left( \frac{dy}{dx} \right)}_{x=4}}.\]
question_answer15) If \[y=\left( 1+x \right)\left( 1+{{x}^{2}} \right)\left( 1+{{x}^{4}} \right)....\left( 1+{{x}^{{{2}^{n}}}} \right)\] then find \[\frac{dy}{dx}\]at\[x=0\].
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