Question 1 :
The set of points where the function $f(x)=x|x|$ is differentiable is?
Question 2 :
Derivative of $\log { \left( \sec { \theta } +\tan { \theta } \right) } $ with respect to $\sec { \theta } $ at $\theta =\cfrac { \pi }{ 4 } $ is ...........
Question 3 :
If $y = a \cos (\log x)$ and $A \frac{d^2y}{dx^2} + B \frac{dy}{dx} + C = 0$, then the values of A,B,C are
Question 4 :
$\dfrac{d}{{dx}}\left[ {\dfrac{{\sin x + \cos x}}{{\sqrt {1 + \sin 2x} }}} \right],\left( {0 < x < \frac{\pi }{4}} \right), = $
Question 6 :
For the function $\displaystyle f\left( x \right) ={ \left( x+1 \right)  }^{ 1/x }$ to be continuous at $x=0$, $f(0)$ must be defined as:
Question 7 :
If $\displaystyle y=\ell n\left \{ \frac{x+\sqrt{(a^{2}+x^{2})}}{a} \right \}$, then the value of $\displaystyle \frac{dy}{dx}$ is
Question 8 :
If $y$ is expressed in terms of a variable $x$ as $y = f(x)$, then $y$ is called
Question 13 :
If $f ( x ) = x ^ { \alpha } \log x$ and $f ( 0 ) = 0 ,$ then the value of $\alpha$ for which $Rolle's theorem$ can be applied in $[ 0,1 ] ,$ is
Question 15 :
If $y = \sqrt {x{{\log }_e}x} $, then $\dfrac{{dy}}{{dx}}$ at $x = e$ is
Question 16 :
Let $f$ be any continuously differentiable function on $[a, b]$ and twice differentiable on $(a, b)$ such that $f(a) = f'(a) = 0$ and $f(b) = 0$. Then
Question 18 :
If $f(x)=\begin{cases}<br>ax &a<1&\\<br>ax^2+bx+2 &a\ge 1&.<br>\end{cases}$<br>Then the values of $a$, $b$ for which $f(x)$ is differentiable, are
Question 19 :
Find the derivative with respect to x of the function$\displaystyle (\log_{\cos x}\sin x)(\log_{\sin x}\cos x)^{-1}+\sin ^{-1}\frac{2x}{1+x^{2}} \mbox{at} x=\frac{\pi }{4}$
Question 20 :
Let $y = x^{x^{x .......}},$ then $\displaystyle \frac{dy}{dx}$ is equal to
Question 21 :
If $f\left( x \right) = \left( {x - a} \right)g\left( x \right)$ and $g\left( x \right)$ is continuous $x = a$ then ${f^{'}}\left( 1 \right) = $
Question 22 :
If $ f\left( x \right) =\begin{cases} 1 & x<0 \\ { x }^{ 2 } & x\ge 0 \end{cases}$ then at $x=0$
Question 23 :
Differential coefficient of $\log (\sin x)$ with respect to $x$ is:
Question 24 :
Let  $f _ { p } ( a ) = e ^ { \frac { i a } { p ^ { 2 } } } \cdot e ^ { \frac { 2 i a } { p ^ { 2 } } } \cdot e ^ { \frac { 3 i a } { p ^ { 2 } } } \ldots \cdot e ^ { \frac { i a } { p } } $  (Where $i = \sqrt { - 1 } \text { and } p \in N ) \text { then } \lim _ { n \rightarrow \infty } f _ { n } ( \pi )$
Question 26 :
Find the values of a and b so that the function $f(x)=\left\{\begin{matrix} x^2+3x+a, & if & x\leq 1\\ bx+2, & if & x > 1\end{matrix}\right.$ is differentiable at each $x\in R$.
Question 27 :
If ${x}^{2}+{y}^{2}=4$, then $y\cfrac { dy }{ dx } +x$ is equal to
Question 28 :
The function $f(x)=x-[x]$, where $[\cdot]$ denotes the greatest integer function is?
Question 30 :
If $\sin ^{ 2 }{ mx } +\cos ^{ 2 }{ ny } ={ a }^{ 2 }$ then $\cfrac{dy}{dx}=$
Question 31 :
Let the equation of a curve is given in implicit form as $y=\tan { \left( x+y \right) } $. Then $\cfrac { { d }^{ 2 }y }{ d{ x }^{ 2 } } $ in terms of $y$ is
Question 32 :
If $y = \tan ^ { - 1 } \left( \cot \left( \dfrac { \pi } { 2 } - x \right) \right) ,$ then $\dfrac { d y } { d x } =$<br/>
Question 33 :
If $y = {\tan ^{ - 1}}\left( {\cot \left( {\frac{\pi }{2} - x} \right)} \right)then\dfrac{{dy}}{{dx}} = $
Question 34 :
If $xy + x^2 y^2 = c$; then the value of $\displaystyle \frac{dy}{dx}$ will be
Question 35 :
If $f(x)=\displaystyle \frac{\sin x}{x},x\neq 0$ is to be continuous at ${x}=0$ then $\mathrm{f}({0})=$<br/>
Question 36 :
If $\displaystyle xe^{xy}-y=\sin x $ then $\displaystyle \frac{dy}{dx}$ at $x = 0$ is
Question 37 :
If the tangent to the curve $x=a \, (\theta + \sin \, \theta), y=a (1+ \cos \,\theta)$ at $ \theta=\dfrac{\pi}{3}$ makes an angle $\alpha (0 \leq\alpha < \pi)$ with x-axis, then $\alpha$ =
Question 43 :
If $y = x - x^{2}$, then the derivatives of $y^{2}$ w.r.t. $x^{2}$ is
Question 44 :
If $\displaystyle x= \sqrt{\frac{1-t^{2}}{1+t^{2}}}$ and $\displaystyle y=\frac{\sqrt{1+t^{2}}-\sqrt{1-t^{2}}}{\sqrt{1+t^{2}}+\sqrt{1-t^{2}}},$ then the value of $\displaystyle \frac{d^{2}y}{dx^{2}}$ at $t= 0$ is given by
Question 45 :
The equation of the tangent to the curve $y=x+\cfrac{4}{{x}^{2}}$, that is parallel to the x-axis is
Question 47 :
Following function is continous at the point $x=2$ <br/>          $f\left( x \right) = \left\{ \begin{array}{l}1 + x,\,\,\,\,when\,\,x < 2\\5 - x,\,\,\,\,when\,\,x \ge 2\end{array} \right.$
Question 50 :
If $f(x)=x\sin x$, then $f' \left ( \dfrac{\pi}{2} \right )$ is equal to
