Derivatives of Composite and Implicit Functions

If $\ln \left(x+y\right)=2xy$, then $y^{\text{'}}\left(0\right)$ is equal to

If $x^2+y^2=\mathrm{1,}$  then :

If the dependent variable y is changed to 'z' by the substitution y = tan z then the differential equation $\frac{{\text{d}}^{2}y}{\text{d}{x}^2}=1+\frac{2\left(1+y\right)}{1+y^2}{\left(\frac{\text{d}y}{\text{d}x}\right)}^2$ is changed to $\frac{{\text{d}}^{2}z}{\text{d}{x}^2}=cos{x}^{2}z+\text{k}{\left(\frac{\text{d}z}{\text{d}x}\right)}^2,$then find the value of k.

Let $P\left(x\right)$ be a polynomial of degree $4$ such that $P\left(1\right)=P\left(3\right)=P\left(5\right)=P^{\text{'}}\left(7\right)=0$. If the real number $x\ne 1,3,5$ is such that $P\left(x\right)=0$ can be expressed as $x=\frac{p}{q}$ where $\text{'}p\text{'} \text{and} \text{'}q\text{'}$ are relatively prime, then $\left(p+q\right)$ equals

Let $f\left(x\right)=x^2-4x-3, x>2$ and let $g\left(x\right)$ be the inverse of $f\left(x\right)$. Find the value of $g^{\text{'}}\left(2\right)$ where $f\left(x\right)=2$

Find the derivative with respect to $x$ of the function :

$\left({\log }_{\cos x}\sin x\right){\left({\log }_{\sin x}\cos x\right)}^{-1}+arc\sin \frac{2x}{1+x^2}$ at $x=\frac{\pi }{4}$

Let$f\left(x\right)=x+\frac{1}{2x+\frac{1}{2x+\frac{1}{2x+\dots \dots \dots \infty }}}$ Then the value of$f\left(100\right)·f^{\text{'}}\left(100\right)$ is 

If $y={\log }_e{\left(x^{e^x·a^y}\right)}^{y^x}$, then $\frac{dy}{dx}$ is equal to

If $y=\frac{x^2}{2}+\frac{1}{2}x\sqrt{x^2+1}+\log \sqrt{x+\sqrt{x^2+1}}$, find the value of $x\frac{dy}{dx}+\log \frac{dy}{dx}$.

If  $y=\frac{{\sin }^{-1}x}{\sqrt{1-x^2}},$ then $\left(1-x^2\right)\frac{d^{2}y}{d{x}^2}-3x\frac{dy}{dx}-y$ is equal to:

$f\left(x\right)=\sqrt{\sin x}$ then $f^{\text{'}}\left(x\right)$ equals to

If $y={\left\{x+\sqrt{x^2+a^2}\right\}}^n$, then $\frac{dy}{dx}$ is

$y={\left({\tan }^{-1}x\right)}^2$, then what would be the value of  ${\left(x^2+1\right)}^2\frac{d^{2}y}{d{x}^2}+2x\left(x^2+1\right)\frac{dy}{dx}$

$\frac{d}{dx}\left(\cos \left(\sin x\right)\right)=$

Let $y=f\left(x\right)$ be a bijective function and $y=g\left(x\right)$ be a function such that $g\left(x\right)=f^{-1}\left(x\right)$. Let $y=h\left(x\right)$ be a function defined as $h\left(x\right)=xf\left(x\right)+g\left(x\right)$. If $y=f\left(x\right)$ and $y=g\left(x\right)$ be differentiable functions, $f\left(1\right)=2$, $f^{\text{'}}\left(1\right)=2$ and $f\left(3\right)=f^{\text{'}}\left(3\right)=1$ then $h^{\text{'}}\left(1\right)=$

If $y=\log \left(x+\sqrt{9+x^2}\right)$, then $\frac{dy}{dx}=$

If $y=6^{e^x\sin x}$ then $\frac{dy}{dx}=$

Differentiate $arc\tan \left(\frac{2+x}{2-x}\right)$