Derivatives of Composite and Implicit Functions

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 $x=\frac{\pi }{4}$

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

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

Find $\frac{d}{dx}\left(\sin \left({\cot }^{-1}x\right)\right)$.

Let $f\left(x\right)=x+\mathrm{c}\mathrm{o}\mathrm{s}x+2$ and $g\left(x\right)$ be the inverse function of $f\left(x\right)$, then $g^{\text{'}}\left(3\right)$ is equal to

Suppose, the function $f\left(x\right)-f\left(2x\right)$ has the derivative $5$ at $x=1$ and derivative $7$ at $x=2$, the derivative of the function $f\left(x\right)-f\left(4x\right)$ at $x=1$ has the value $10+\lambda$, then the value of $\lambda$ is equal to:

If $y=\frac{\left(4-x\right)\sqrt{4-x}-\left(6-x\right)\sqrt{x-6}}{\sqrt{4-x}+\sqrt{x-6}}$, then $\frac{dy}{dx}$ is equal to

Differentiate the following w.r.t $x:$

${\cos }^{-1}\left\{\sqrt{\frac{1+x}{2}}\right\}$

The derivative of the function ${\cos }^{-1}\left\{\sqrt{\frac{1+x}{2}}\right\}$ is

Differentiate ${\tan }^{-1}\sqrt{\frac{1-\sin 4x}{1+\sin 4x}}$ w.r.t $x$.

If $y={\tan }^2\left(\log x^3\right),$ then $\frac{dy}{dx}$ is

Differentiation of $\log (\sec x+\tan x)$ w.r.t. $x$ is

Differentiation of $\sin \left(x^2+x\right)$ w.r.t. $x$ is