Embibe Experts Solutions for Chapter: Differentiation, Exercise 1: Exercise 1

The derivative of $\sin x^3$ with respect to $\cos x^3$ is equal to

If $y=\sqrt{x+\sqrt{y+\sqrt{x+\sqrt{y+\mathrm{.....}\infty }}}}$, then $\frac{dy}{dx}$ is equal to

If $x{e}^{xy}=y+{\sin }^{2}x,$ then $\frac{dy}{dx}$ at $x=0$ is

If $f\left(x\right)=\frac{{\cos }^{2}x}{1+{\sin }^{2}x}$, then $f\left(\frac{\pi }{4}\right)-3f^{\text{'}}\left(\frac{\pi }{4}\right)=$

If $y={\log }_e{\left(\frac{1+x}{1-x}\right)}^{\frac{1}{4}}-\frac{1}{2}{\tan }^{-1}\left(x\right),$ then $\frac{dy}{dx}$ at $x=\frac{1}{\sqrt{2}}$ equals $\_\_\_\_\_\_\_\_\_$

If $y=\frac{1}{x}+\cos 2x$, then $\frac{d^{2}y}{d{x}^2}=$

If $y=f\left(x^3\right), z=g\left(x^5\right), f^{\text{'}}\left(x\right)=\tan x$ and $g^{\text{'}}\left(x\right)=\sec x$, then the value of $\frac{dy}{dz}$ is

$\underset{x\to \frac{\pi }{2}}{\lim }\left(\frac{\sin x-{\left(\sin x\right)}^{\sin x}}{1-\sin x+\ln \left(\sin x\right)}\right)=$