Basics of Differentiation

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

if y = y(x) and it follows the relation $e^{xy}+ycosx=\mathrm{2,}$ then find (i) y' (0) and (ii) y'' (0)

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 $f\left(x\right)=2$

Suppose $f\left(x\right)=\tan \left({\sin }^{-1}\left(2x\right)\right)$, then $f^{\text{'}}\left(\frac{1}{4}\right)$ is equal to

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 $y={\log }_e{\left(x^{e^x·a^y}\right)}^{y^x}$, 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 $x\frac{dy}{dx}+\log \frac{dy}{dx}$.

$f\left(x\right)=\sqrt{\sin x}$ 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  ${\left(x^2+1\right)}^2\frac{d^{2}y}{d{x}^2}+2x\left(x^2+1\right)\frac{dy}{dx}$

On differentiating  $\sqrt{\tan x}$ with respect to $x$ from first principle, the solution would be:

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

Let $h=fog,$ where $f : R\to R$ and $g : R\to R$ are differentiable functions,

If $f\left(2\right)=1, f^{\text{'}}(-1)=-2, f^{\text{'}}\left(4\right)=3, g\left(2\right)=-1, g^{\text{'}}\left(2\right)=4, g^{\text{'}}\left(1\right)=8,$ then $h^{\text{'}}\left(2\right)=$

If $e^{\frac{x}{y}}-{\tan }^{-1}\left(\frac{y}{x}\right)=2020$ for $x\ne 0$ and $y\ne 0,$ we get $\frac{dy}{dx}=$