If slope of tangent of curve $y=\dfrac{x}{b-x}$ at $(1,1)$ be $2$ then $b=$

The general solution of the differential equation $\sin{2x}\left( \cfrac { dy }{ dx } -\sqrt { \tan { x }  }  \right) -y=0$ is $y\phi(x)=x+c$ then ${ \Phi  }^{ 1 }\left( \cfrac { \pi  }{ 4 }  \right) $ is _____

The derivative of a differentiable even function is always an even function.

If $g$ is the inverse of $f$ and $\displaystyle f'\left( x \right) =\frac { 1 }{ 1+{ x }^{ 3 } } $, then $g'\left( x \right) $ is equal to

If $y=mx+c$ is the normal at a point $(8,8)$  on the parabola ${ y }^{ 2 }=8x$ Find $m$

Function $ f(x)= \sin^{-1} (3x-4x^3) $ is-

If $\log \sqrt{x^2+y^2}=\tan^{-1}\left(\dfrac{y}{x}\right)$ , then $\dfrac{dy}{dx}$ is:

Average Speed and Instantaneous speed are the same things.

If $y = \left| {\cos x} \right| + \left| {\sin x} \right|$ , then ${\dfrac{dy} {dx}}$ at $x = {\dfrac {2\pi } 3}$ is

The line $y =\sqrt{2}x + 4\sqrt{2}$ is a normal to $y^{2} =4ax$ then a = 

The  number of points where $f(x) = \mid | x |^2 - 5| x | + 6 \mid $ is non-derivable is/are

Let $f(x) = \left{\begin{matrix} 2 + 1,& x \leq 1\ x^{2} + 2, & 1 < x \leq 2\ 4x - 2, & x > 2\end{matrix}\right.$ then the number of points where $f(x)$ is non-differentiable, is equal to

$f(x)=|\cos x|$ is not differentiable for the points given by $x=?$

If $g$ is the inverse of $f$ and $f'(x)=\dfrac{1}{1+x^{3}}$, then $g'(x)$ is equal to.

If $f(x)=(x^2-4)\left|x^3-6x^2+11x-6\right|+\dfrac{x}{1+|x|}$, then the set of points at which the function $f(x)$ is not differentiable is?

If $\sqrt { { x }^{ 2 }+{ y }^{ 2 } } ={ e }^{ t }$ where $t=\sin ^{ -1 }{ \left( \cfrac { y }{ \sqrt { { x }^{ 2 }+{ y }^{ 2 } }  }  \right)  } $ then $\cfrac { dy }{ dx } $ is equal to

Value of c is :-
$\dfrac{d}{dx}(c\ ^{f(x)}) = f' (x)e^{f(x)}$

The condition that the line $\dfrac{x}{a}+\dfrac{y}{b}=1$ is tangent to the curve $x^{2/3}+y^{2/3}=1$ is

If line $PQ$, whose equation is $y = 2x + k,$  is a normal to the parabola whose vertex is   $(-2,3)$ and the axis parallel to the $x$-axis with latus rectum equal to $2$, then the possible value of k is

If $y = \dfrac { 1 } { 1 + x ^ { n - m } + x ^ { p - m } } + \dfrac { 1 } { 1 + x ^ { m - n } + x ^ { p - n } } + \dfrac { 1 } { 1 + x ^ { m - p } + x ^ { n - p } }$ then $\dfrac { d y } { d x }$ at $x = e ^ { m ^ { n p } }$ is equal to

Let f(x) be a differentiable function and $f\left( \alpha  \right) = f\left( \beta  \right) = 0\,\left( {\alpha  < \beta } \right)$, then in the interval $\left( {\alpha ,\beta } \right)$

If $y = \log \left( \frac { 1 + x } { 1 - x } \right) ^ { 1 / 4 } - \frac { 1 } { 2 } \tan ^ { - 1 } x ,$ then $\frac { d y } { d x } =$

If f'$\left( x \right) =\sqrt { { 2x }^{ 2 }-1 } $ and y=f$\left( { x }^{ 2 } \right) $ then $\dfrac { dy }{ dx } $ at x=1 is

A differentiable function function $y = h(x)$ satisfies $\displaystyle \overset{x}{\underset{0}{\int}} (x - t + 1)h(t)dt = x^4 + x^2; \forall x \ge 0$, then value of $h(0) + h'(0)$ is equal to

Area of the triangle formed by the lines $x-y=0, x+y=0$ and ant tangent to the hyparabola $x^{2}-y^{2}=a^{2}$ is 

If $x = \exp \left{ \tan ^ { - 1 } \left( \frac { y - x ^ { 2 } } { x ^ { 2 } } \right) \right}$ then $\frac { d y } { d x } =$

Consider the function $f(x)=\mathrm{s}\mathrm{g}\mathrm{n} x$ and $g(x)=x\left ( 1-x^{2} \right )$. Which of the following does NOT hold good?

The set of all points of differentiability of the function $\displaystyle f(x) = \dfrac{\sqrt{x + 1} 1}{\sqrt{x}}$ for $x$  and $f(0)$ = 0 is

If the graph of the equation $y = 2x^2 - 6x + C$ is tangent to the $x$-axis, the value of $C$ is

$f\left( x \right) =\begin{cases} x;\quad x<1 \ 3-x;\quad 1\le x\le 3 \end{cases}$ then $f^{'}(x)=$

The domain of the derivative of the function
$\displaystyle f\left ( x \right )=\begin{cases}
\tan^{-1}x & \text{ if } \left | x \right |\leq 1 \
\frac{1}{2}\left ( \left | x \right |-1 \right ) & \text{ if } \left | x \right |> 1
\end{cases}$

Let $f(x)=ax^2+bx+c$ such that $f(1)=f(-1)$ and a, b, c are in Arithmetic Progression.

If $y=\displaystyle\dfrac{1}{a-z}$, then $\displaystyle\dfrac{dz}{dy}$ is:

Which of the following given statements is/are correct?

The value of $\displaystyle \frac{d}{dx} (|x-1|+ |x-5|) $ at x = 3 is

Let $f : R \rightarrow R$ be a function defined by $f(x)= \max\left { x,  x^3 \right }$. The set of all points where $f(x)$ is NOT differentiable is:

If $f(x) = \left{\begin{matrix}e^x+ax & x< 0 \ b(x-1)^2 & x \geq 0 \end{matrix}\right.$ is differentiable at $x= 0$, then $(a, b)$ is

If $f(x) =x[x \sqrt{x}-\sqrt{x+1}]$, then:

If $y = \dfrac {1}{1 + x^{n - m} + x^{p - m}} + \dfrac {1}{1 + x^{m - n} + x^{p - n}} + \dfrac {1}{1 + x^{m - p} +x^{n - p}}$ then $\dfrac {dy}{dx}$ at $e^{m^{n^{p}}}$ is equal to

If the distance between a tangent to the parabola $y^{2} = 4x$ and a parallel normal to the same parabola is $2\sqrt{2}$, then possible values of gradient of either of them are:

Consider the function $f(x)=\begin{cases} x^2 \sin \dfrac{1}{x};x\neq 0 \ 0 ; otherwise  \end{cases}$
then,

Consider the following statements:
$1.$ Derivative of $f(x)$ may not exist at some point.
$2.$ Derivative of $f(x)$ may exist finitely at some point.
$3.$ Derivative of $f(x)$ may be infinite (geometrically) at some point.
Which of the above statements are correct?

A function is defined in $(0, \infty)$ by
$f(x) = \left{\begin{matrix}1 - x^{2} & for & 0 < x  \leq 1\ \ln\ x & for & 1 < x \leq 2\ \ln\ 2 - 1 + 0.5x & for & 2 < x < \infty\end{matrix}\right.$
Which one of the following is correct in respect of the derivative of the function, i.e., $f'(x)$?

 lf $\mathrm{f}(\mathrm{x})=\left{\begin{array}{l}1, \mathrm{x}<0\1+ \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{x}, 0\leq \mathrm{x}</\pi _{2} \end{array}\right.$, then derivative of f(x) at$\mathrm{x}=0$

If $f(x)=(4+x)^{n}$,$n \epsilon N$ and $f^{r}(0)$ represents the $r^{th}$ derivative of f(x) at x = 0, then the value of $\sum _{r=0}^{\infty}\frac{(f^{r}(0))}{r!}$ is equal to

Let $f(x)=\begin{cases}\begin{matrix} 1 & \forall &  x<0 \ 1+\sin x & \forall & 0\leq x\leq \dfrac{\pi}2\end{matrix}\end{cases}$ then what is the value of $f'(x)$ at $x=0?$

The second order differential equation is :

$f\left( x \right) = \left| {x - 1} \right| + \left| {x + 2} \right| + \left| {x - 3} \right|$ is not differentiable at