Rules of Differentiation

If $y={\tan }^{-1}\left(\frac{a_{1}x-\alpha }{a_1\alpha +x}\right)+{\tan }^{-1}\left(\frac{a_2-a_1}{1+a_1{a}_2}\right)+{\tan }^{-1}\left(\frac{a_3-a_2}{1+a_2{a}_3}\right)+....+{\tan }^{-1}\left(\frac{a_n-a_{n-1}}{1+a_n{a}_{n-1}}\right)-{\tan }^{-1}\left(a_n\right)$, then find $\frac{dy}{dx}$.

If $y=\left(1+x\right)\left(1+x^2\right)\left(1+x^4\right)... \left(1+x^{2n}\right),$ find $\frac{dy}{dx} at x=0$.

If $f^{\text{'}}\left(x\right)=\sqrt{2x^2-1}$ and $y=f\left(x^2\right),$ then $\frac{dy}{dx}$ at $x=-1,$ is

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

If $y=f\left(\frac{3x+4}{5x+6}\right)$ and $f^{\text{'}}\left(x\right)=\tan x^2$, then $\frac{dy}{dx}$ is equal to

If $f\left(x\right)=\cos x\cos 2x\cos 4x\cos 8x\cos 16x$, then $f\text{'}\left(\frac{\pi }{4}\right)$ is

Let $f\left(x\right)=\sin x, g\left(x\right)=x^2$ and $h\left(x\right)={\log }_{e}x$. If $F\left(x\right)=h\left(f\left(g\left(x\right)\right)\right)$, then $F^{\text{'}}\left(x\right)$ is

If $f\left(x\right)={\log }_e\left|x\right|, x\ne 0,$ then $f^{\text{'}}\left(x\right)$ is equals to

The differential coefficient of $f\left({\log }_{e}x\right)$ w.r.t. $x,$ where $f\left(x\right)={\log }_{e}x$, is:

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

$\sqrt{4+\sqrt{4+\sqrt{4+x^2}}}$

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

$\frac{\sqrt{a^2+x^2}+\sqrt{a^2-x^2}}{\sqrt{a^2+x^2}-\sqrt{a^2-x^2}}$

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

$5^{\left(3-x^2\right)}+{\left(3-x^2\right)}^5$

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

${\log }_{10}x+{\log }_{x}10+{\log }_{x}x+{\log }_{10}10$ 

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

$\frac{x{\sin }^{-1}x}{\sqrt{1-x^2}}+{\log }_e\left(\sqrt{1-x^2}\right), \left|x\right|<1$

Differentiate the following w.r.t. $x$: $e^{{\cos }^{-1}\left(\sqrt{1-x^2}\right)}, \left|x\right|<1$.

Differentiate the following w.r.t. $x$: $a^{{\left({\sin }^{-1}x\right)}^2}, \left|x\right|<1$.

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

$\sin \left(m{\sin }^{-1}\left(x\right)\right), \left|x\right|<1$.

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

$\frac{e^x+{\log }_e\left(x\right)}{\sin 3x}$

Differentiate the following w.r.t. $x$: ${\log }_e\left(\sqrt{\frac{1+\sin x}{1-\sin x}}\right)$.

If  $x\sqrt{1+y}+y\sqrt{1+x}=0$, then prove that $\frac{dy}{dx}=\frac{-1}{{\displaystyle {\left(x+1\right)}^2}}$.