Differentiation practice · Easy

A particle is moving along the parabola $y^2=4(x+2)$. As it passes through the point $(7,6)$ its $y$-coordinate is increasing at the rate of 3 units per second. The rate at which $x$-coordinate change at this instant is (in units/sec)

The derivative of $f(x)=3|2+x|$ at the point $x_{0}=-3$ is

Let $f$ be a bijection satisfying $f^{\prime}(x)=f(x)$. Then, $\left(f^{-1}\right)^{\prime \prime}(x)$ is equal to

Let $f: R \rightarrow R$ be a function given by $f(x+y)=f(x) f(y)$ for all $x, y \in R$

If $f(x) \neq 0$ for all $x \in R$ and $f^{\prime}(0)$ exists, then $f^{\prime}(x)$ equals

If $f(x+y)=f(x) \cdot f(y) \forall x \cdot y$ and $f(5)=2$, $f^{\prime}(0)=3$, then $f^{\prime}(5)$ is

If $f(x)=\log |2 x|, x \neq 0$ then $f^{\prime}(x)$ is equal to

$\frac{d}{d x} \cos ^{-1} \frac{x-x^{-1}}{x+x^{-1}}$ is equal to

If $f\left( x \right) = {x^n},$ then the value of $f\left( 1 \right) - \frac{{f'\left( 1 \right)}}{{1!}} + \frac{{f''\left( 1 \right)}}{{2!}} - \frac{{f'''\left( 1 \right)}}{{3!}} + \ldots + \frac{{{{\left( { - 1} \right)}^n}{f^n}\left( 1 \right)}}{{n!}}$ is

Let $f$ be a twice differentiable function such that $f''\left( x \right) = - f\left( x \right)$ and $f'\left( x \right) = \mathrm{g}\left( x \right).$ If $h'\left( x \right) = \left[ {f{{\left( x \right)}^2} + \mathrm{g}{{\left( x \right)}^2}} \right]h\left( 1 \right) = 8$ and $h\left( 0 \right) = 2,$ then $h\left( 2 \right)$ is equal to

Which of the following is/are true?

I. The derivative of $\sin (x+a)$ is $\cos (x+a)$, where $a$ is a fixed non-zero constant.

II. The derivative of $\operatorname{cosec} x \cot x$ is $\operatorname{cosec}^3 x-\cot ^2 x \operatorname{cosec} x$

If $y = {5^x}{x^5},$, then $\frac{{dy}}{{dx}}$ is

If $f\left( x \right)$ and $g\left( x \right)$ are two functions with $g\left( x \right) = x - \frac{1}{x}$ and $fo\mathrm{g}\left( x \right) = {x^3} - \frac{1}{{{x^3}}},$ then $f'\left( x \right)$ is

If $y=(\sin x)^{\tan x}$, then $\frac{d y}{d x}$ is equal to

If $y = \frac{{\sec x - \tan x}}{{\sec x + \tan x}},$ then $\frac{{dy}}{{dx}}$ equals

If $f(x)=\sin ^{-1}\left(2 x \sqrt{1-x^2}\right), x \in[-1,1]$ Then,

If $G\left( x \right) = \sqrt {25 - {x^2}}$, then $\lim\limits_{x \to 1} \frac{{G\left( x \right) - G\left( 1 \right)}}{{x - 1}}$ has the value

Let $f(x) = \left\{ {\begin{array}{cccccccccccccccccccc} {\frac{{\sin (x - [x])}}{{x - [x]}},}&{x \in ( - 2, - 1)} \\ {\max \,\,\,\left\{ {2x,3\left[ {\left| x \right|} \right]} \right\},}&{\left| x \right| < 1} \\ {1,}&{{\text{other}}\,\,{\text{wise}}} \end{array}} \right.$
Where $[t]$ denotes greatest integer $\leqslant t.$ If $m$ is the number of points where $f$ is not continuous and $n$ is the number of points where $f$ is not differentiable, then the ordered pair $(m,n)$ is

If ${y^2} = a{x^2} + bx + c,$ where $a,b,c$ are constants,then ${y^3}\frac{{{d^2}y}}{{d{x^2}}}$ is equal to

If $e^{x-y}=x^y$, then $\frac{d y}{d x}$ is equal to

The set of all points where the function $f(x)=x|x|$ is twice differentiable is

If the greatest value of $\frac{x^2-x+c}{x^2+x+c}$ is $\frac{5}{3}$, then find the value of $c$.

If $f(x)=\sin \left\{\frac{\pi}{3}[x]-x^2\right\}$ for $2 < x < 3$ and $[x]$ denotes the greatest integer less than or equal to $x$, then $f^{\prime}(\sqrt{\pi / 3})$ is equal to

$f\left( x \right) = \left| {\begin{array}{cccccccccccccccccccc} {{x^3}}&{{x^4}}&{3{x^2}}\\ 1&{ - 6}&4\\ p&{{p^2}}&{{p^3}} \end{array}} \right|$, here $p$ is a constant, then $\frac{{{d^4}f\left( x \right)}}{{d{x^4}}}$ is

If ${x^x}{y^y}{z^z} = c$, then $\frac{{\partial z}}{{\partial x}}$ is equal to

If $f(x)=e^x g(x), g(0)=2, g^{\prime}(0)=1$, then $f^{\prime}(0)$ is

If $y = \sin [{\cos ^{ - 1}}\{ \sin ({\cos ^{ - 1}}x)\} ],$ then $\frac{{dy}}{{dx}}$ at $x = \frac{1}{2}$ is equal to

Let $f$ be the function on $[0,1]$ given by $f(x)=$ $x \sin \frac{\pi}{x}$ for $x \neq 0$ and $f(0)=0$. Then $f$ is

If $y=\cos ^{-1} \frac{9-x^2}{9+x^2}$ then $y^{\prime}(-1)$ is equal to

If $y = {\sin ^{ - 1}}\frac{x}{2} + {\cos ^{ - 1}}\frac{x}{2},$ then the value of $\frac{{dy}}{{dx}}$ is

Let $f(x)=x^2+p x+q$. The value of $(p, q)$ so that $f(1)=3$ is an extreme value of $f$ on $[0,2]$ is