12th Class Mathematics Applications of Derivatives Question Bank

done Case Based (MCQs) - Derivatives Total Questions - 40

• question_answer1)

  Directions: (1 - 5)

  Let \[f\text{ }\left( x \right)\]be a real valued function, then its

  Left Hand Derivative (L.H.D.) :

  \[Lf'\left( a \right)=\underset{h\to 0}{\mathop{\lim }}\,\frac{f\left( a-h \right)-f\left( a \right)}{-h}\]

  Right Hand Derivative (R.H.D.) :

  \[Rf'\left( a \right)=\underset{h\to 0}{\mathop{\lim }}\,\frac{f\left( a+h \right)-f\left( a \right)}{h}\]

  Also, a function \[f\left( x \right)\]is said to be differentiable at x = a if its L.H.D. and R.H.D. at x = a exist and are equal.

  For the function  answer the following questions

  R.H.D. of \[f\left( x \right)\] at \[x=1\] is
  A)
  1 done clear
  B)
  -1 done clear
  C)
  0 done clear
  D)
  2 done clear
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• question_answer2)
  L.H.D. of \[f\left( x \right)\] at \[x=1\] is
  A)
  1 done clear
  B)
  -1 done clear
  C)
  0 done clear
  D)
  2 done clear
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• question_answer3)
  \[f\left( x \right)\] is non-differentiable at
  A)
  x = 1 done clear
  B)
  x = 2 done clear
  C)
  x = 3 done clear
  D)
  x = 4 done clear
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• question_answer4)
  Find the value of\[f'\left( 2 \right)\]
  A)
  1 done clear
  B)
  2 done clear
  C)
  3 done clear
  D)
  -1 done clear
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• question_answer5)
  The value of f''(-1) is
  A)
  2 done clear
  B)
  1 done clear
  C)
  -2 done clear
  D)
  -1 done clear
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• question_answer6)

  Directions: (6 - 10)

  Let \[f\left( x \right)\text{ }=\text{ }f\left( t \right)\] and \[y=g\left( t \right)\] be parametric forms with t as a parameter, then \[\frac{dy}{dx}=\frac{dy}{dt}\times \frac{dt}{dx}=\frac{g'\left( t \right)}{f'\left( t \right)}\], where \[f'\left( t \right)\ne 0\].

  On the basis of above information, answer the following questions.

  The derivative of \[f\left( \tan \,x \right)\]w.r.t. \[g\left( sec\text{ }x \right)\]at \[x=\frac{\pi }{4}\], where \[f'\left( 1 \right)=2\] and \[g'\left( \sqrt{2} \right)=4\], is
  A)
  \[\frac{1}{\sqrt{2}}\] done clear
  B)
  \[\sqrt{2}\] done clear
  C)
  1 done clear
  D)
  0 done clear
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• question_answer7)
  The derivative of \[{{\sin }^{-1}}\left( \frac{2x}{1+{{x}^{2}}} \right)\]with respect to \[{{\cos }^{-1}}\left( \frac{1-{{x}^{2}}}{1+{{x}^{2}}} \right)\] is
  A)
  - 1 done clear
  B)
  1 done clear
  C)
  2 done clear
  D)
  4 done clear
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• question_answer8)
  The derivative of \[{{e}^{{{x}^{3}}}}\]with respect to log x is
  A)
  \[{{e}^{{{x}^{3}}}}\] done clear
  B)
  \[3{{x}^{2}}\,2{{e}^{{{x}^{3}}}}\] done clear
  C)
  \[3{{x}^{3}}{{e}^{{{x}^{3}}}}\] done clear
  D)
  \[3{{x}^{2}}{{e}^{{{x}^{3}}}}+3x\] done clear
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• question_answer9)
  The derivative of \[{{\cos }^{-1}}\left( 2x-1 \right)\]w.r.t. \[{{\cos }^{-1}}x\]is
  A)
  2 done clear
  B)
  \[\frac{-1}{2\sqrt{1-{{x}^{2}}}}\] done clear
  C)
  \[\frac{2}{x}\] done clear
  D)
  \[1-{{x}^{2}}\] done clear
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• question_answer10)
  If \[y=\frac{1}{4}\,{{u}^{4}}\] and \[u=\frac{2}{3}\,{{x}^{3}}+5\], then \[\frac{dy}{dx}=\]
  A)
  \[\frac{2}{27}{{x}^{2}}{{\left( 2{{x}^{3}}+15 \right)}^{3}}\] done clear
  B)
  \[\frac{2}{27}{{x}^{2}}{{\left( 2{{x}^{3}}+15 \right)}^{3}}\] done clear
  C)
  \[\frac{2}{27}x{{\left( 2{{x}^{3}}+5 \right)}^{3}}\] done clear
  D)
  \[\frac{2}{27}{{\left( 2{{x}^{3}}+15 \right)}^{3}}\] done clear
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• question_answer11)

  Directions: (11 - 15)

  Let \[f\,\,:\,\,A\to B\] and \[g\,\,:\,\,B\to C\] be two functions defined on non-empty sets A, B, C, then gof : \[A\to C\] be is called the composition of f and g defined as, \[gof\left( x \right)=\,g\left\{ f\left( x \right) \right\}\forall x\in A\].

  Consider the funciton  \[g\left( x \right)={{e}^{x}}\] and then answer the following questions.

  The function \[gof\left( x \right)\] is defined as
  A)
  done clear
  B)
  done clear
  C)
  done clear
  D)
  done clear
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• question_answer12)
  \[\frac{d}{dx}\left\{ gof\left( x \right) \right\}=\]
  A)
  done clear
  B)
  done clear
  C)
  done clear
  D)
  done clear
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• question_answer13)
  R.H.D. of \[gof\left( x \right)\]at x = 0 is
  A)
  0 done clear
  B)
  1 done clear
  C)
  -1 done clear
  D)
  2 done clear
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• question_answer14)
  L.H.D. of \[gof\left( x \right)\]at x = 0 is
  A)
  0 done clear
  B)
  1 done clear
  C)
  - 1 done clear
  D)
  2 done clear
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• question_answer15)
  The value of \[f'\left( x \right)\]at \[x=\frac{\pi }{4}\] is
  A)
  1/9 done clear
  B)
  \[1/\sqrt{2}\] done clear
  C)
  1/2 done clear
  D)
  not defined done clear
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• question_answer16)

  Directions: (16 - 20)

  If a real valued function \[f\left( x \right)\]is finitely derivable at any point of its domain, it is necessarily continuous at that point. But its converse need not be true.

  For example, every polynomial, constant function are both continuous as well as differentiable and inverse trigonometric functions are continuous and differentiable in its domains etc.

  Based on the above information, answer the following questions.

   then at x = 0
  A)
  \[f\left( x \right)\] is differentiable and continuous done clear
  B)
  \[f\left( x \right)\] is neither continuous nor differentiable done clear
  C)
  \[f\left( x \right)\] is continuous but not differentiable done clear
  D)
  none of these done clear
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• question_answer17)
  If \[f\left( x \right)=\left| x-1 \right|,\,\,x\in R\], then at x=1
  A)
  \[f\left( x \right)\]is not continuous done clear
  B)
  \[f\left( x \right)\] is continuous but not differentiable done clear
  C)
  \[f\left( x \right)\] is continuous and differentiable done clear
  D)
  none of these done clear
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• question_answer18)
  \[f\left( x \right)={{x}^{3}}\] is
  A)
  continuous but not differentiable at x = 3 done clear
  B)
  continuous and differentiable at x = 3 done clear
  C)
  neither continuous nor differentiable at x = 3 done clear
  D)
  none of these done clear
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• question_answer19)
  If \[f\left( x \right)=\left[ sin\text{ }x \right]\], then which of the following is true?
  A)
  \[f\left( x \right)\]is continuous and differentiable at x = 0. done clear
  B)
  \[f\left( x \right)\]is discontinuous at x = 0 done clear
  C)
  \[f\left( x \right)\]is discontinuous at x = 0 but not differentiable. done clear
  D)
  \[f\left( x \right)\] is differentiable but not continuous at \[x=\pi /2\]. done clear
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• question_answer20)
  If\[f\left( x \right)={{\sin }^{-1}}x,\,-1\,\le \,x\,\le \,\,1\], then
  A)
  \[f\left( x \right)\] is both continuous and differentiable done clear
  B)
  \[f\left( x \right)\] is neither continuous nor differentiable. done clear
  C)
  \[f\left( x \right)\] is continuous but not differentiable. done clear
  D)
  None of these. done clear
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• question_answer21)

  Directions: (21 - 25)

  Derivative of \[y\text{ }=\text{ }f\left( x \right)\]w.r.t. x (if exists) is denoted by \[\frac{dy}{dx}\] of \[f'\left( x \right)\]and is called the first order derivative of y.

  If we take derivative of again, then we get \[\frac{dy}{dx}\left( \frac{dy}{dx} \right)=\frac{{{d}^{2}}y}{d{{x}^{2}}}\] or \[f''\left( x \right)\] and is called the second order derivative of y. Similarly, \[\frac{d}{dx}\left( \frac{{{d}^{2}}y}{d{{x}^{2}}} \right)\] is denoted and defined as \[\frac{{{d}^{3}}y}{d{{x}^{3}}}\]or \[f'''\left( x \right)\] and is known as third order derivative of y and so on.

  Based on the above information, answer the following questions.

  If \[y={{\tan }^{-1}}\left( \frac{\log \left( e/{{x}^{2}} \right)}{\log \left( e{{x}^{2}} \right)} \right)+{{\tan }^{-1}}\left( \frac{3+2\log \,x}{1-6\log \,x} \right)\], then \[\frac{{{d}^{2}}y}{d{{s}^{2}}}\]is equal to
  A)
  2 done clear
  B)
  1 done clear
  C)
  0 done clear
  D)
  -1 done clear
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• question_answer22)
  If \[u={{x}^{2}}+{{y}^{2}}\] and \[x=s+3t\], \[y=2s-t\], then \[\frac{{{d}^{2}}u}{d{{s}^{2}}}\] is equal to
  A)
  12 done clear
  B)
  32 done clear
  C)
  36 done clear
  D)
  10 done clear
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• question_answer23)
  \[f\left( x \right)=2\,\,\log \,\,\sin \,x\], then \[f''\left( x \right)\] is equal to
  A)
  \[2\,\,\cos e{{c}^{3}}\,x\] done clear
  B)
  \[2\,{{\cot }^{2}}x-4{{x}^{2}}\,\cos e{{c}^{2}}\,{{x}^{2}}\] done clear
  C)
  \[2x\,\cot \,{{x}^{2}}\] done clear
  D)
  \[-2\,\cos e{{c}^{2}}\,x\] done clear
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• question_answer24)
  If \[f\left( x \right)={{e}^{x}}\,\sin \,x\], then \[f'''\left( x \right)=\]
  A)
  \[2{{e}^{x}}\left( \sin \,x+\cos \,x \right)\] done clear
  B)
  \[2{{e}^{x}}\left( \cos x-\sin x \right)\] done clear
  C)
  \[2{{e}^{x}}\left( \sin \,x-\cos \,x \right)\] done clear
  D)
  \[2{{e}^{x}}\,\cos x\] done clear
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• question_answer25)
  If \[{{y}^{2}}=a{{x}^{2}}+bx+c\], then \[\frac{d}{dx}\left( {{y}^{3}}{{y}_{2}} \right)=\]
  A)
  1 done clear
  B)
  -1 done clear
  C)
  \[\frac{4ac-{{b}^{2}}}{{{a}^{2}}}\] done clear
  D)
  0 done clear
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• question_answer26)

  Directions: (26 - 30)

  Logarithmic differentiation is a powerful technique to differentiate functions of the form \[f\left( x \right)={{\left[ u\left( x \right) \right]}^{v\left( x \right)}}\], where both u(x) and v(x) are differentiable functions and f and u need to be positive functions.

  Let   function   \[y=f\left( x \right)={{\left( u\left( x \right) \right)}^{v\left( x \right)}}\],   then

  \[y'=y\left[ \frac{v(x)}{u(x)}u'(x)+v'(x)\centerdot log[u(x)] \right]\]

  On the basis of above information, answer the following questions.

  Differentiate \[{{x}^{x}}\] w.r.t. x
  A)
  \[{{x}^{x}}\left( 1+\log \,x \right)\] done clear
  B)
  \[{{x}^{x}}\left( 1-\log \,x \right)\] done clear
  C)
  \[-{{x}^{x}}\left( 1+\log \,x \right)\] done clear
  D)
  \[{{x}^{x}}\,\log \,x\] done clear
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• question_answer27)
  Differentiate \[{{x}^{x}}+{{a}^{x}}+{{x}^{a}}+{{a}^{a}}\] w.r.t. x
  A)
  \[\left( 1+\log \,x \right)+\left( {{a}^{x}}\,\log \,a+a{{x}^{a-1}} \right)\] done clear
  B)
  \[{{x}^{x}}\left( 1+\log \,x \right)+\log \,a+a{{x}^{a-1}}\] done clear
  C)
  \[{{x}^{x}}\left( 1+\log \,x \right)+{{x}^{a}}\,\operatorname{logx}+a{{x}^{a-1}}\] done clear
  D)
  \[{{x}^{x}}\left( 1+\log \,x \right)+{{a}^{x}}\,\log \,a\,+\,a{{x}^{a-1}}\] done clear
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• question_answer28)
  If \[x={{e}^{x/y}}\], then find \[\frac{dy}{dx}\].
  A)
  \[-\frac{\left( x+y \right)}{x\,\log \,x}\] done clear
  B)
  \[-\frac{\left( x-y \right)}{x\,\log \,x}\] done clear
  C)
  \[\frac{\left( x+y \right)}{x\,\log \,x}\] done clear
  D)
  \[\frac{\left( x-y \right)}{x\,\log \,x}\] done clear
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• question_answer29)
  If \[y={{\left( 2-x \right)}^{3}}\,{{\left( 3+2x \right)}^{5}}\], then find \[\frac{dy}{dx}\].
  A)
  \[{{\left( 2-x \right)}^{3}}{{\left( 3+2x \right)}^{5}}\left[ \frac{15}{3+2x}-\frac{8}{2-x} \right]\] done clear
  B)
  \[{{\left( 2-x \right)}^{3}}{{\left( 3+2x \right)}^{5}}\left[ \frac{15}{3+2x}+\frac{3}{2-x} \right]\] done clear
  C)
  \[{{\left( 2-x \right)}^{3}}{{\left( 3+2x \right)}^{5}}\left[ \frac{10}{3+2x}-\frac{3}{2-x} \right]\] done clear
  D)
  \[{{\left( 2-x \right)}^{3}}{{\left( 3+2x \right)}^{5}}\left[ \frac{10}{3+2x}+\frac{3}{2-x} \right]\] done clear
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• question_answer30)
  If \[y={{x}^{x}}\,\,{{e}^{\left( 2x+5 \right)}}\] , then \[\frac{dy}{dx}\]is
  A)
  \[{{x}^{x}}{{e}^{\left( 2x+5 \right)}}\,\left( 2+\log \,\,x \right)\] done clear
  B)
  \[{{x}^{x}}{{e}^{\left( 2x+5 \right)}}\,\left( 3+2\,\log \,x \right)\] done clear
  C)
  \[{{x}^{x}}\,{{e}^{\left( 2x+5 \right)}}\,\,\left( 2+3\,\log \,x \right)\] done clear
  D)
  \[{{x}^{x}}\,{{e}^{\left( 2x+5 \right)}}\,\left( 3+\log \,x \right)\] done clear
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• question_answer31)

  Directions: (31 - 35)

  If \[y=f\left( u \right)\]is a differentiable function of u and \[u=g\left( x \right)\]is a differentiable function of x, then \[y=f\left[ \left( g\left( x \right) \right. \right]\]is a differentiable function of x and\[\frac{dy}{dx}=\frac{dy}{du}\times \frac{du}{dx}\]. This rule is also known as CHAIN RULE.

  Based on the above information, find the derivative of functions w.r.t. x in the following questions.

  \[\cos \sqrt{x}\]
  A)
  \[\frac{-\sin \sqrt{x}}{2\sqrt{x}}\] done clear
  B)
  \[\frac{\sin \sqrt{x}}{-\sin \sqrt{x}}\] done clear
  C)
  \[sin\sqrt{x}\] done clear
  D)
  \[-\sin \sqrt{x}\] done clear
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• question_answer32)
  \[{{7}^{x+\frac{1}{x}}}\]
  A)
  \[\left( \frac{{{x}^{2}}-1}{{{x}^{2}}} \right)\centerdot {{7}^{x+\frac{1}{x}}}\centerdot \log 7\] done clear
  B)
  \[\left( \frac{{{x}^{2}}+1}{{{x}^{2}}} \right)\centerdot {{7}^{x+\frac{1}{x}}}\centerdot \log 7\] done clear
  C)
  \[\left( \frac{{{x}^{2}}-1}{{{x}^{2}}} \right)\centerdot {{7}^{x-\frac{1}{x}}}\centerdot \log 7\] done clear
  D)
  \[\left( \frac{{{x}^{2}}+1}{{{x}^{2}}} \right)\centerdot {{7}^{x+\frac{1}{x}}}\centerdot \log 7\] done clear
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• question_answer33)
  \[\sqrt{\frac{1-\cos x}{1+\cos \,x}}\]
  A)
  \[\frac{1}{2}{{\sec }^{2}}\frac{x}{2}\] done clear
  B)
  \[-\frac{1}{2}{{\sec }^{2}}\frac{x}{2}\] done clear
  C)
  \[{{\sec }^{2}}\frac{x}{2}\] done clear
  D)
  \[-{{\sec }^{2}}\frac{x}{2}\] done clear
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• question_answer34)
  \[\frac{1}{b}{{\tan }^{-1}}\left( \frac{x}{b} \right)+\frac{1}{a}{{\tan }^{-1}}\left( \frac{x}{a} \right)\]
  A)
  \[\frac{-1}{{{x}^{2}}+{{b}^{2}}}+\frac{1}{{{x}^{2}}+{{a}^{2}}}\] done clear
  B)
  \[\frac{1}{{{x}^{2}}+{{b}^{2}}}+\frac{1}{{{x}^{2}}+{{a}^{2}}}\] done clear
  C)
  \[\frac{1}{{{x}^{2}}+{{b}^{2}}}-\frac{1}{{{x}^{2}}+{{a}^{2}}}\] done clear
  D)
  none of these done clear
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• question_answer35)
  \[{{\sec }^{-1}}x+\cos e{{c}^{-1}}\frac{x}{\sqrt{{{x}^{2}}-1}}\]
  A)
  \[\frac{2}{\sqrt{{{x}^{2}}-1}}\] done clear
  B)
  \[\frac{-2}{\sqrt{{{x}^{2}}-1}}\] done clear
  C)
  \[\frac{1}{\left| x \right|\,\sqrt{{{x}^{2}}-1}}\] done clear
  D)
  \[\frac{2}{\left| x \right|\,\sqrt{{{x}^{2}}-1}}\] done clear
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• question_answer36)

  Directions: (36 - 40)

  If a relation between x and y is such that y cannot be expressed in terms of x, then y is called an implicit function of x. When a given relation expresses y as an implicit function of x and we want to find \[\frac{dy}{dx}\], then we differentiate every term of the given relation w.r.t x, remembering that a term in y is first differentiated w.r.t. y and then multiplied by \[\frac{dy}{dx}\].

  Based on the above information, find the value of \[\frac{dy}{dx}\]in each of the following questions.

  \[{{x}^{3}}+{{x}^{2}}y+x{{y}^{2}}+{{y}^{3}}=81\]
  A)
  \[\frac{\left( 3{{x}^{2}}+2xy+{{y}^{2}} \right)}{{{x}^{2}}+2xy+3{{y}^{2}}}\] done clear
  B)
  \[\frac{-\left( 3{{x}^{2}}+2xy+{{y}^{2}} \right)}{{{x}^{2}}+2xy+3{{y}^{2}}}\] done clear
  C)
  \[\frac{\left( 3{{x}^{2}}+2xy-{{y}^{2}} \right)}{{{x}^{2}}-2xy+3{{y}^{2}}}\] done clear
  D)
  \[\frac{3{{x}^{2}}+xy+{{y}^{2}}}{{{x}^{2}}+xy+3{{y}^{2}}}\] done clear
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• question_answer37)
  \[{{x}^{y}}={{e}^{x-y}}\]
  A)
  \[\frac{x-y}{\left( 1+\log \,x \right)}\] done clear
  B)
  \[\frac{x+y}{\left( 1+\log \,x \right)}\] done clear
  C)
  \[\frac{x-y}{x\left( 1+\log x \right)}\] done clear
  D)
  \[\frac{x+y}{x\left( 1+\log \,x \right)}\] done clear
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• question_answer38)
  \[{{e}^{\sin \,y}}=xy\]
  A)
  \[\frac{-y}{x\,\left( y\,\cos y-1 \right)}\] done clear
  B)
  \[\frac{y}{v\,\cos \,y-1}\] done clear
  C)
  \[\frac{y}{y\,cos\,y+1}\] done clear
  D)
  \[\frac{y}{x\left( y\,\cos y-1 \right)}\] done clear
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• question_answer39)
  \[{{\sin }^{2}}x+{{\cos }^{2}}y=1\]
  A)
  \[\frac{\sin \,2y}{\sin \,2x}\] done clear
  B)
  \[-\frac{\sin \,2x}{\sin \,2y}\] done clear
  C)
  \[-\frac{\sin \,2y}{\sin \,2x}\] done clear
  D)
  \[\frac{\sin \,2y}{\sin \,2x}\] done clear
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• question_answer40)
  \[y={{\left( \sqrt{x} \right)}^{{{\sqrt{x}}^{\sqrt{x}....\infty }}}}\]
  A)
  \[\frac{-{{y}^{2}}}{x\left( 2-y\,\log \,x \right)}\] done clear
  B)
  \[\frac{{{y}^{2}}}{2+y\,\log \,x}\] done clear
  C)
  \[\frac{{{y}^{2}}}{x\left( 2+y\,\log \,x \right)}\] done clear
  D)
  \[\frac{{{y}^{2}}}{x\left( 2-y\,\log \,x \right)}\] done clear
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