CBSE Questions for Class 11 Commerce Applied Mathematics Differentiation Quiz 3 - MCQExams.com

lf $$\mathrm{y}=\log(\log(x+\sqrt{1+x^{2}}))$$ then $$\displaystyle \frac{dy}{dx}=$$
  • $$\displaystyle \frac{1}{x+\sqrt{1+x^{2}}}$$
  • $$\displaystyle \frac{x}{\log(x+\sqrt{1+x^{2}})}$$
  • $$\displaystyle \frac{-1}{\log(x+\sqrt{1+x^{2}})}$$
  • $$\displaystyle \frac{1}{\sqrt{1+x^{2}}\log(x+\sqrt{1+x^{2}})}$$
lf $$\mathrm{a}^{\mathrm{y}}=\log_{\mathrm{a}}(\mathrm{x}^{2}+\mathrm{x}+1)$$, then $$\displaystyle \frac{dy}{dx}=$$
  • $$\displaystyle \frac{\log_{a}e.(2x+1)}{(x^{2}+x+1)\log(x^{2}+x+1)}$$
  • $$\displaystyle \frac{(2x+1)}{(x^{2}+x+1)\log(x^{2}+x+1)}$$
  • $$\displaystyle \frac{1}{(x^{2}+x+1)\log(x^{2}+x+1)}$$
  • $$\displaystyle \frac{-1}{(x^{2}+x+1)\log(x^{2}+x+1)}$$
If $$\sqrt{x+y}+\sqrt{y-x}=c$$, then $$\dfrac{d^2y}{dx^2}$$ is
  • $$\displaystyle \frac{2}{c}$$
  • $$\displaystyle -\frac{2}{c^2}$$
  • $$\displaystyle \frac{2}{c^2}$$
  • $$\displaystyle -\frac{2}{c}$$
If $$f(x)=(ax+b)\cos x + (cx+d)\sin x$$ and $$f^{'}(x)=x \cos x$$, for all values of $$x\in R$$, then $$a,b,c,d$$ are given by
  • $$a = b = c = d$$
  • $$0, 1, -1, 0$$
  • $$1, 0, -1, 0$$
  • $$0, 1, 1, 0$$
The set onto which the derivative of the function $$f(x)=x(\log x -1)$$ maps the ray $$[1, \infty)$$ is ?
  • $$[1, \infty)$$
  • $$(10, \infty)$$
  • $$[0,\infty)$$
  • $$(0,0)$$
lf $$y=\displaystyle \frac{\log(x+\sqrt{1+x^{2}})}{\sqrt{1+x^{2}}}$$ then $$(1+x^{2})y_{1}+xy=$$
  • $$y$$
  • $$-y$$
  • $$0$$
  • $$1$$
lf $$f(x)=\displaystyle \frac{x^{2}}{x+a}$$ then $$f^{'}(a)=$$
  • 4
  • $$\displaystyle \frac{3}{8}$$
  • $$\displaystyle \frac{3}{4}$$
  • 8
If $${ e }^{ y }+xy=e,$$ then $$\displaystyle{ \frac { { d }^{ 2 }y }{ d{ x }^{ 2 } }  }_{ x=0 }^{  }$$ is
  • $$\displaystyle \frac { 1 }{ { e }^{ 2 } } $$
  • $${e}^{-1}$$
  • $$e$$
  • None of these

If $$x^{4}+y^{4}-a^{2}xy=0$$ defines $${y}$$ implicitly as function of $$x$$ , then $$ \displaystyle \frac{dy}{dx}=$$
  • $$\displaystyle \frac{4x^{3}-a^{2}y}{4y^{3}-a^{2}x}$$
  • $$-\left(\displaystyle \frac{4x^{3}-a^{2}y}{4y^{3}-a^{2}x}\right)$$
  • $$\displaystyle \frac{4x^{3}}{4y^{3}-a^{2}x}$$
  • $$\displaystyle \frac{-4x^{3}}{4y^{3}-a^{2}x}$$

 lf $$\mathrm{y}=(\mathrm{x}^{2}+1)^{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{x}}$$, then $$\mathrm{y}^{'}(0)$$ is equal to
  • $$\dfrac{1}{2}$$
  • $$e^{2}$$
  • $$0$$
  • $$\dfrac{3}{2}$$
If $$f(x)=e$$ $$^{x}$$ $$g(x),$$
$$g(0)=1,g'(0)=3$$, then$$ f' (0)$$ is
  • $$0$$
  • $$4$$
  • $$3$$
  • $$7$$

 If $$ax^{2}+2hxy+by^{2}+2gx+2fy +c=0$$ then $$\displaystyle \frac{dy}{dx}=$$
  • $$-(\displaystyle \frac{ax+hy+g}{hx+by+f})$$
  • $$-(\displaystyle \frac{ax+hy+g}{bx+hy+f})$$
  • $$-(\displaystyle \frac{hx+by+f}{ax+hy+g})$$
  • $$-(\displaystyle \frac{hx+by+f}{hx+ay+g})$$
If $$u=\displaystyle \tan^{-1}\left (\frac{x^{2}+y^{2}}{x+y}\right)$$, then $$x\displaystyle \frac{d u}{d x}+y\frac{d u}{d y}=$$
  • $$\sin 2u$$
  • $$\dfrac {1}{2}\sin 2u$$
  • $$\dfrac {1}{3}\sin 2u$$
  • $$2\sin 2u$$
If $$y=\sqrt{\sin{x}+y}$$, then $$\displaystyle\frac{dy}{dx}$$ is
  • $$\displaystyle\frac{\cos{x}}{2y-1}$$
  • $$\displaystyle\frac{\cos{x}}{1-2y}$$
  • $$\displaystyle\frac{\cos{x}}{2y+1}$$
  • $$\displaystyle\frac{\sin{x}}{2y-1}$$
If $$\sqrt{x}+\sqrt{y}=4$$, then find $$\displaystyle\frac{dx}{dy}$$ at $$y=1$$.
  • $$3$$
  • $$4$$
  • $$-3$$
  • $$-4$$
If $$x^3+y^3=3axy$$, then $$\displaystyle\frac{dy}{dx}$$ is
  • $$\displaystyle\frac{ay-y^2}{x^2-ax}$$
  • $$\displaystyle\frac{ay-x^2}{y^2-ax}$$
  • $$\displaystyle\frac{by-x^2}{y^2-bx}$$
  • None of these
For the function $$f(x) = \displaystyle \frac{x^{100}}{100} + \frac{x^{99}}{99} + ........... + \frac{x^2}{2} + x+1$$, $$f'(1) =$$
  • $$x^{100}$$
  • $$100$$
  • $$101$$
  • None of these
If $$ y=x+\displaystyle\frac{1}{x+\displaystyle\frac{1}{x+\displaystyle\frac{1}{x+\cdots}}}$$ then $$\displaystyle\frac{dy}{dx}=$$
  • $$\displaystyle\frac{dy}{dx}=\frac{x}{2x-y}$$
  • $$\displaystyle\frac{dy}{dx}=\frac{y}{2y-x}$$
  • $$\displaystyle\frac{dy}{dx}=\frac{2y}{2x-y}$$
  • None of these
  • Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
  • Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
  • Assertion is correct but Reason is incorrect
  • Assertion is incorrect but Reason is correct
If $$\displaystyle x^2+y^2=t-\frac{1}{t}$$ and $$\displaystyle x^4+y^4=t^2+\frac{1}{t^2}$$, then $$\displaystyle x^3y\frac{dy}{dx}=$$
  • $$0$$
  • $$1$$
  • $$-1$$
  • none of these
If $$xy+y^2=\tan x + y$$, then $$\displaystyle\frac{dy}{dx}$$ is equal to
  • $$\displaystyle\frac{\sec^{2}{x}-y}{(x+2y-1)}$$
  • $$\displaystyle\frac{\cos^{2}{x}+y}{(x+2y-1)}$$
  • $$\displaystyle\frac{\sec^{2}{x}-y}{(2x+y-1)}$$
  • $$\displaystyle\frac{\cos^{2}{x}+y}{(2x+2y-1)}$$
If $$y^x=x^y$$, then find $$\displaystyle\frac{dy}{dx}$$.
  • $$\displaystyle\frac{x(y\log{y}-y)}{y(x\log{x}-x)}$$
  • $$\displaystyle\frac{y(x\log{y}-y)}{x(y\log{x}-x)}$$
  • $${y(x\log{y}-y)}$$
  • $$\displaystyle\frac{y(x\log{x}-y)}{x(y\log{y}-x)}$$
Differentiate $$\sqrt{\displaystyle\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}}$$ with respect to $$x$$.
  • $$\displaystyle\frac{1}{2}\sqrt{\displaystyle\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}}\left[\frac{1}{x-1}-\frac{1}{x-2}-\frac{1}{x-3}-\frac{1}{x-4}-\frac{1}{x-5}\right]$$
  • $$\displaystyle\frac{1}{2}\sqrt{\displaystyle\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}}\left[\frac{1}{x-2}+\frac{1}{x-3}+\frac{1}{x-4}+\frac{1}{x-5}\right]$$
  • $$\displaystyle\frac{1}{2}\sqrt{\displaystyle\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}}\left[\frac{1}{x-1}+\frac{1}{x-2}-\frac{1}{x-3}-\frac{1}{x-4}-\frac{1}{x-5}\right]$$
  • None of these
If $$y=x^{-\tfrac12}+\log_5x+\displaystyle \frac {\sin x}{\cos x}+2^x$$, then find $$\dfrac {dy}{dx}$$
  • $$-\displaystyle \frac {1}{2}x^{-3/2}+\displaystyle \frac {1}{x\log_e5}+\sec^2x+2^x\log 2$$
  • $$\displaystyle \frac {1}{2}x^{-3/2}+\displaystyle \frac {1}{x\log_e5}+\sec^2x+2^x\log 2$$
  • $$-\displaystyle \frac {3}{2}x^{-3/2}+\displaystyle \frac {1}{x\log_e5}+\sec^2x+2^x\log 2$$
  • $$-\displaystyle \frac {1}{2}x^{-3/2}+\displaystyle \frac {1}{x\log_e5}+\cos^2x+2^x\log 2$$
If $$y=e^x \sin x$$, then find $$\displaystyle \frac {dy}{dx}$$
  • $$e^x(\sin x+\cos x)$$
  • $$e^x(\sin x-\cos x)$$
  • $$e^x \sin x$$
  • None of these
$$\displaystyle\frac{dy}{dx}$$ for $$y=x^x$$ is
  • $$x^x(1-\log{x})$$
  • $$x^x(1-\log{y})$$
  • $$x^x(1+\log{y})$$
  • $$x^x(1+\log{x})$$
If $$y=x^2+sin^{-1}x+log_ex$$, find $$\dfrac {dy}{dx}$$
  • $$\displaystyle \frac {dy}{dx}=2x+\displaystyle \frac {1}{\sqrt {1-x^2}}+\displaystyle \frac {1}{x}$$
  • $$\displaystyle \frac {dy}{dx}=x+\displaystyle \frac{1}{\sqrt {1-x^2}}+\displaystyle \frac {1}{x}$$
  • $$\displaystyle \frac {dy}{dx}=2x+\displaystyle \frac {1}{\sqrt {1-x^2}}-\displaystyle \frac {1}{x}$$
  • $$\frac {dy}{dx}=2x-\displaystyle \frac {1}{\sqrt {1-x^2}}+\displaystyle \frac {1}{x}$$
If $$y=\sqrt{\sin{x}+\sqrt{\sin{x}+\sqrt{\sin{x}+\cdots\:to\:\infty}}}$$, then $$\displaystyle\frac{dy}{dx}$$ is
  • $$\displaystyle\frac{\cos{x}}{1+2y}$$
  • $$\displaystyle -\frac{\sin{x}}{1-2y}$$
  • $$\displaystyle\frac{\cos{x}}{1-2y}$$
  • $$\displaystyle\frac{\cos{x}}{2y-1}$$
If $$y=e^x \tan x + x\cdot \log_ex$$, then find $$\displaystyle \frac {dy}{dx}$$
  • $$\displaystyle \frac {dy}{dx}=e^x \tan x+(\log x+1)$$
  • $$\displaystyle \frac {dy}{dx}=e^x(\tan x+\sec^2x)+(\log x+x)$$
  • $$\displaystyle \frac {dy}{dx}=e^x \tan x+1$$
  • $$\displaystyle \frac {dy}{dx}=e^x(\tan x+\sec^2x)+(\log x+1)$$
If $$y={(\tan{x})}^{\displaystyle{(\tan{x})}^{\displaystyle\tan{x}}}$$, then find $$\displaystyle\frac{dy}{dx}$$ at $$\displaystyle x=\frac{\pi}{4}$$.
  • $$0$$
  • $$1$$
  • $$-1$$
  • $$2$$
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Practice Class 11 Commerce Applied Mathematics Quiz Questions and Answers