If $y={\tan }^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right),$then $y^{\prime }\left(0\right)$ is

Important Questions on Continuity and Differentiability

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

Let $f$ and $g$ be differentiable functions such that $f\left(3\right)=5, g\left(3\right)=7,f^{\prime }\left(3\right)=13,g^{\prime }\left(3\right)=6,f^{\prime }\left(7\right)=2$ and $g^{\prime }\left(7\right)=0$$.$ If $h\left(x\right)=\left(fog\right)\left(x\right),$ then $h^{\prime }\left(3\right)=$

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

If for $x\in \left(0,\frac{1}{4}\right),$ the derivative of ${\tan }^{-1}\left(\frac{6x\sqrt{x}}{1-9x^3}\right)$ is $\sqrt{x} \cdot g\left(x\right)$ , then $g\left(x\right)$ equals:

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

If $f$ is differentiable at $x=1$ and $f\left(1\right)=0,$ $\underset{h\to 0}{\lim }\frac{1}{h}f(1+h)=5,$ then $f^{\prime }(1)=$

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

Let $f:f\left(-x\right)\to f\left(x\right)$ be a differentiable function. If $f$ is even, then $f^{\prime }\left(0\right)$ is equal to

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

If $y=\sec \left({\tan }^{-1}x\right)$, then $\frac{dy}{dx}$ at $x=1$ is equal to

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

If $y=\frac{{\sin }^{2}x}{1+\cot x}+\frac{{\cos }^{2}x}{1+\tan x}$$,$ then $y^{\prime }\left(x\right)$ is equal to

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

If $g\left(x\right)$ is the inverse function of $f\left(x\right)$ and $f^{\text{'}}\left(x\right)=\frac{1}{1+x^4},$ then $g^{\text{'}}\left(x\right)$ is

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

If $\int e^{\sec x}\left(\sec x\tan xf\left(x\right)+\left(\sec x\tan x+{\sec }^{2}x\right)\right)dx=e^{\sec x}f\left(x\right)+C,$ then a possible choice of $f\left(x\right)$ is:

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

If $\sin y=x\sin \left(a+y\right)$, then $\frac{dy}{dx}$ is

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

If the function $f\left(x\right)$ defined by $f\left(x\right)=\frac{x^{100}}{100}+\frac{x^{99}}{99}+......+\frac{x^2}{2}+x+1,$ then $f^{\prime }\left(0\right)=$

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

Let $f$ be a polynomial function such that $f\left(3x\right)=f^{\prime }\left(x\right). f^{\prime \prime }\left(x\right),$ for all $x\in R.$ Then :

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

For $x>1$, if ${\left(2x\right)}^{2y}=4e^{2x-2y}$, then ${\left(1+{\log }_{e}2x\right)}^2 \frac{dy}{dx}$ is equal to

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

Let $p\left(x\right)$ be a polynomial such that $p\left(x\right)-p^{\text{'}}\left(x\right)=x^n$ where $n$ is a positive integer. Then, $p\left(0\right)$ equals

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

If $f\left(x\right)=x^6+6^x,$ then $f^{\prime }\left(x\right)$ is equal to

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

If $f\left(1\right)=1,f^{\text{'}}\left(1\right)=3$, then the derivative of $f\left(f\left(f\left(x\right)\right)\right)+{\left(f\left(x\right)\right)}^2$ at $x=1$ is:

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

If $y\left(\alpha \right)=\sqrt{2\left(\frac{\mathrm{t}\mathrm{a}\mathrm{n}\alpha +\mathrm{c}\mathrm{o}\mathrm{t}\alpha }{1+\mathrm{t}\mathrm{a}{\mathrm{n}}^2\alpha }\right)+\frac{1}{\mathrm{s}\mathrm{i}{\mathrm{n}}^2\alpha }}, \alpha \in \left(\frac{3\pi }{4},\pi \right)$, then $\frac{dy}{d\alpha }$ at $\alpha =\frac{5\pi }{6}$ is

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

Let $f\left(x\right)={\log }_e\left(\mathrm{s}\mathrm{i}\mathrm{n}x\right),\mathrm{ }\left(0<x<\pi \right)$ and $g\left(x\right)={\sin }^{-1}\left(e^{-x}\right),\mathrm{ }(x\ge 0).$ If $\alpha$ is a positive real number such that $a={\left(fog\right)}^{\mathrm{\text{'}}}\left(\alpha \right)$ and $b=\left(fog\right)\left(\alpha \right),$ then

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

If $y={\sin }^{-1}\frac{1}{2}\left(\sqrt{1+x}+\sqrt{1-x}\right)$ then $y^{\prime }$ is equal to

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

Let $f$ be a differentiable function such that $f\left(1\right)=2$ and $f^{\text{'}}\left(x\right)=f\left(x\right)$ for all $x\in R$. If $h\left(x\right)=f\left(f\left(x\right)\right)$, then $h^{\text{'}}\left(1\right)$ is equal to :

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Derivatives of Composite and Implicit Functions

If $y=\frac{x+c}{1+x^2}$ where $c$ is a constant, when $y$ is stationary, $xy$ is equal to -