If $x^2+y^2=\mathrm{1,}$  then :

Important Questions on Continuity and Differentiability

EASY

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

If then $\mathrm{}{x}^y=e^{x-y},$ then $\frac{dy}{dx}$ at $x=1$ is ….

MEDIUM

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

If $g$ is the inverse of a function $f$ and $f^{\text{'}}\left(x\right)=\frac{1}{1+x^5},$ then $g^{\text{'}}\left(x\right)$ is equal to 

EASY

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

If $e^y+xy=e,$ the ordered pair $\left(\frac{dy}{dx},\frac{d^{2}y}{d{x}^2}\right)$ at $x=0$ is equal to

MEDIUM

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

If $\sqrt[3]{y}\sqrt{x}=\sqrt[6]{{\left(x+y\right)}^5},$ then $\frac{dy}{dx}=$

MEDIUM

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

If $x^m{y}^n={\left(x+y\right)}^{m+n}$ then $y^{\prime }$ is equal to

MEDIUM

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

If $x=2\sin \theta -\sin 2\theta$ and $y=2\cos \theta -\cos 2\theta ,$$\theta \in \left[0,2\pi \right],$ then $\frac{d^{2}y}{d{x}^2}$ at $\theta =\pi$ is:

HARD

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

The derivative of ${\mathrm{t}\mathrm{a}\mathrm{n}}^{-1}\left(\frac{\mathrm{s}\mathrm{i}\mathrm{n}x-\mathrm{c}\mathrm{o}\mathrm{s}x}{\mathrm{s}\mathrm{i}\mathrm{n}x+\mathrm{c}\mathrm{o}\mathrm{s}x}\right)$ with respect to $\frac{x}{2},$ where $x\in \left(0,\frac{\pi }{2}\right)$, is

HARD

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

The derivative of ${\tan }^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)$ with respect to ${\tan }^{-1}\left(\frac{2x\sqrt{1-x^2}}{1-2x^2}\right)$ at $x=\frac{1}{2}$ is :

MEDIUM

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

If $x=\sqrt{2^{{\mathrm{cosec}}^{-1}t}}$ and  $y=\sqrt{2^{{\sec }^{-1}t}}, \left(\left|t\right|\ge 1\right),$ then $\frac{dy}{dx}$ is equal to

MEDIUM

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

If $x=f\left(t\right)$ and $y=g\left(t\right)$ are differentiable functions of $t$ then $\frac{d^{2}y}{d{x}^2}$ is

MEDIUM

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

If $x^3+2xy+\frac{1}{3}{y}^3=\frac{11}{3}$, then $\frac{dy}{dx}$ at $(2,-1)$ is

MEDIUM

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

If $x=3 \mathrm{t}\mathrm{a}\mathrm{n}t$ and $y=3 \mathrm{s}\mathrm{e}\mathrm{c}t,$ then the value of $\frac{d^{2}y}{d{x}^2}$ at $t=\frac{\pi }{4},$ is:

MEDIUM

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

If $y=(\cos x{)}^{2x}$, then $\frac{dy}{dx}$ is equal to

EASY

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

If $x=\mathrm{s}\mathrm{i}\mathrm{n}\theta , y={\mathrm{s}\mathrm{i}\mathrm{n}}^3\theta ,$ then $\frac{d^{2}y}{d{x}^2}$ at $\theta$ = $\frac{\pi }{2}$ is ….

MEDIUM

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

Suppose a differentiable function $f\left(x\right)$ satisfies the identity $f\left(x+y\right)=f\left(x\right)+f\left(y\right)+x{y}^2+x^{2}y,$ for all real $x$ and $y$. If $\underset{x\to 0}{\lim }\frac{f\left(x\right)}{x}=1,$ then $f^{\text{'}}\left(3\right)$ is equal to :

EASY

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

The derivative of ${\cos }^{-1}\left(2x^2-1\right)$ w.r.t., ${\cos }^{-1}x$ is

MEDIUM

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

If $x^m y^n=2{\left(x+y\right)}^{m+n}$, the value of $\frac{dy}{dx}$ is

MEDIUM

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

If $x=a\left(t-\frac{1}{t}\right), y=a\left(t+\frac{1}{t}\right)$ where $t$ be the parameter then $\frac{dy}{dx}=?$

MEDIUM

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

If $y=x^{x^{x^{.....\infty }}},$ then $y^{\prime }$ is equal to

EASY

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

Let $f\left(-x\right)=f\left(x\right)$. Then $f^{\text{'}}\left(x\right)$ must be