Differentiation using Chain Rule or Substitution

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 $x\frac{dy}{dx}+\log \frac{dy}{dx}$.

Let $f\left(x\right)=\left|\ln x\right|, x\ne 1$

What is the derivative of $f\circ f\left(x\right)$, where $1<x<2$?

Differentiate:

$(ax+b{)}^n(cx+d{)}^m$

Find the derivative of ${\sin }^{n}x$

If $f\left(x\right)={\log }_e\left({\log }_{e}x\right),$ then $f^{\text{'}}\left(e\right)$ is equal to

$\frac{d}{dx}\left(2\sin \frac{7x}{12}\right)=$

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

Let $f\left(x\right)$ be a one-to-one function such that $f\left(1\right)=3, f\left(3\right)=1, f^{\text{'}}\left(1\right)=-4$ and $f^{\text{'}}\left(3\right)=2$. If $g=f^{-1}$, then slope of the tangent line to $\frac{1}{g}$ at $x=1$ is:

Let $f\left(x\right)=e^{x^3+x^2+x}$ & $g$ is inverse of $f\left(x\right)$ then find $g^{\text{'}}\left(e^3\right)$?

If $f\left(x\right)={\log }_e\left(e^{2x}{\left(\frac{3x+5}{5-3x}\right)}^{\frac{2}{3}}\right), x\ne \frac{-5}{3},\frac{5}{3}$, then the value of $\frac{df}{dx}$ at $x=1$, is

If $f\left(x\right)=\sin \left(\cosh \left(\frac{x^2+1}{x^2+2}\right)\right)$ then $f^{\text{'}}\left(1\right)=$

If $y=e^{3\log \left(2x+1\right)}$, then $\frac{dy}{dx}=$

If $f\left(x\right)=x^3+x-5$ and $g\left(x\right)$ be another function such that $g\left(f\left(x\right)\right)=x$ then find the value of $g^{\text{'}}\left(63\right)$

If $y={\sin }^{-1}\left(\frac{2^{x+1}}{1+4^x}\right)$ and $\frac{dy}{dx}=\frac{2^{x+1}\log 2}{f\left(x\right)}$ then $f\left(0\right)=\_\_\_\_\_$.

$\frac{d}{dx}\left[\log \left(\frac{1}{x}\right)+\log \left(\frac{1}{x^2}\right)+\log \left(\frac{1}{x^3}\right)\right]=$_____ $x>1$