Question 2 :
If $y=x-x^2$, then the derivative of $y^2$ with respect to $x^2$ is
Question 3 :
Let $g$ is the inverse function of $f$ and $f'\left( x \right) = {{{x^{10}}} \over {\left( {1 + {x^2}} \right)}}$.If $g\left( 2 \right) = a$, then $g'\left( 2 \right)$ is equal to
Question 4 :
Let $u(x)$ and $v(x)$ are differentiable functions such that $\displaystyle \frac{u(x)}{v(x)}=7$.If $\displaystyle \frac{u^{'}(x)}{v^{'}(x)} = p$ and $\left ( \displaystyle \frac{u(x)}{v(x)} \right )^{'} =q$ then $\left ( \displaystyle \frac{p+q}{p-q} \right )$ has the value equal to
Question 5 :
Let $f,g$ and $h$ are differentiable functions. If $f(0)=1;g(0)=2;h(0)=3$ and the derivative of their pair wise products at $x=0$ are $(fg)'(0)=6;(gh)'(0)=4$ and $(hf)'(0)=5$ then compute the value of $(fgh)'(0).$
Question 6 :
Let $f(x)=x^n$, $n$ being a non - negative integer . The number of values of $n$ for which ${f}'(p+q)={f}'(p)+{f}'(q)$ is valid for all $p,q>0$ is :
Question 7 :
If $u=f(x^3), v=g(x^2), f'(x)=cos x$ and $g'(x)=sin x$, then find $\displaystyle \frac {du}{dv}$
Question 9 :
$\displaystyle f\left( x \right)=\frac { { x }^{ 2 }-x }{ { x }^{ 2 }+2x } $, then $\displaystyle \frac { d{ f }^{ -1 }\left( x \right)  }{ dx } $ is equal to
Question 10 :
If $\mathrm{g}$ is the inverse of $\mathrm{f}$ and $\displaystyle \mathrm{f'}(\mathrm{x})=\frac{1}{2+\mathrm{x}^{\mathrm{n}}}$ , then $\mathrm{g'}(\mathrm{x})$<br/>
Question 11 :
If $g$ is inverse function of $f$ where <br>$f(x)=\int _{ 0 }^{ \pi }{ \cfrac { 1 }{ \sqrt { 1+{ t }^{ 2 } } } } dt\quad $ and<br>$\int { g{ \left( g'(x) \right) }^{ 2 }dx } =\cfrac { { \left[ 1+{ \left( g(x) \right) }^{ \alpha } \right] }^{ \beta } }{ \gamma } +c$. Then the value of $\alpha \beta \gamma $ is equal to [where $\alpha ,\beta ,\gamma \in R$]
Question 12 :
If $f(x) = \cos^2x + \cos^2\left(x + \displaystyle\frac{\pi}{3}\right) + \sin x\sin\left(x + \displaystyle\frac{\pi}{3}\right)$ and <br>$\\ g\left(\displaystyle\frac{5}{4}\right) = 3$, then $\displaystyle\frac{d}{dx}\left(gof(x)\right)$
Question 14 :
If $g$ is the inverse function of $f$ and $f^{'}(x)=\sin x$ then $g^{'}(x)$ is
Question 15 :
Let $\displaystyle F\left( x \right) ={ \left( f\left( \frac { x }{ 2 }  \right)  \right)  }^{ 2 }+{ \left( g\left( \frac { x }{ 2 }  \right)  \right)  }^{ 2 },F\left( 5 \right) =5$ and $f''\left( x \right) =-f\left( x \right),g\left( x \right) =f'\left( x \right) $, then $F(10)$ is equal to
Question 16 :
Let $f$ and $g$ be two differentiable functions satisfying $g(a)=b$, $g'(a)=2$ and $fog=I$ <span>(where $I$ is the identity function). Then $f'(b)$ is equal to?</span>
Question 17 :
If $y=fofof\left( x \right) $ and $f\left( 0 \right) =0,{ f }^{ \prime }\left( 0 \right) =2$, then find ${ y }^{ \prime }\left( 0 \right) $
Question 18 :
If $f(x) = \displaystyle \sin \left ( \frac{\pi }{2}\left [ x \right ]-x^{5} \right )1< x< 2$ and $\left [ x \right ]$ denotes the greatest integer less than or equal to $x$, then $\displaystyle f'\left ( \sqrt[5]{\frac{\pi }{2}} \right )$ is equal to
Question 19 :
If f and g are inverse of each other then which of the following statement is <b>CORRECT</b> ?<br>$I.$ if $f(x) = x^{3} + 2e^{x}$ then ${g}'(2) =\frac{1}{2}.$<br>$II.$ if $f(x) = e^{x}-e^{-x}$ then ${g}'(2)=\frac{1}{2\sqrt{2}}.$<br>$III.$ if $f(x) = x^{3} +2x+2$ then ${g}'(2)=\frac{1}{2}$<br><br>
Question 21 :
Let $\mathrm{g}(\mathrm{x})$ be the inverse of the function $\mathrm{f}(\mathrm{x})$ and $\displaystyle \mathrm{f}'(\mathrm{x})=\frac{1}{1+\mathrm{x}^{3}}$ Then $\mathrm{g'}(\mathrm{x})$ is<br>
Question 24 :
If $y=f(x)=x^3+x^5$ and g is the inverse of f, then find $g'(2)$ <br>
Question 25 :
If $y=\log { \sqrt { \tan { x }  }  } $, then the value of $\cfrac{dy}{dx}$ at $x=\cfrac { \pi  }{ 4 } $ is given by
Question 27 :
Let $f$ be a bijection satisfying $f'(x) = f(x)$. Then, $(f^{-1})''(x)$ is equal to
Question 28 :
$\frac{d}{{dx}}\left( {\frac{{1 + {x^2} + {x^4}}}{{1 + x + {x^2}}}} \right) = ax + b$, then $\left( {a,b} \right) = $
Question 32 :
If $y = \dfrac{{\sin \left( {x + 9} \right)}}{{\cos x}},\,then\,\dfrac{{dy}}{{dx}}\,at\,x = 0$, is
Question 33 :
If f is one-one and satisfies $ \displaystyle {f}'\left ( x \right )=\sqrt{1-\left ( f\left ( x \right ) \right )^{2}} $, then $ \displaystyle {\left ( f^{-1} \right )}'\left ( x \right ) $<br>
Question 34 :
If $y = f\left(\displaystyle\frac{2x - 1}{x^2 + 1}\right)$ and $f'(x) = \sin x^2$, then $\displaystyle\frac{dy}{dx}$ is equal to
Question 35 :
If $ g$ is the inverse of a function $ f$ and $ f'(x)=\displaystyle \dfrac{1}{1+x^{5}}$, then $ g'(x)$ is equal to
Question 36 :
If f(x)= sinx +sin4x-cosx, then $f'(2x^{2}+\frac{\pi }{2})$ at $x=\sqrt{\frac{\pi }{2}}$ is equal to 
Question 37 :
If $f(0)=0$, ${f}'\left ( 0 \right )=2$ then the derivative of $y=f(f(f(f(x))))$ at $x=0$ is<br>
Question 38 :
If $\displaystyle f(x)=x^2-x+5, x>\frac {1}{2}$, and $g(x)$ is its inverse function, then $g'(7)$ equals :
Question 39 : Let $f(x)=[x]$ and<br/>$g\left ( x \right )=\begin{cases}<br/>0 & \text{ if } x\:is\:an\:integer \\ <br/>x^{2} & otherwise <br/>\end{cases}$ <div>then<br/><br/></div>
