Derivatives of Composite and Implicit Functions

If $x^2+y^2=\mathrm{1,}$  then :

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 $x$.

Find $\frac{dy}{dx}$, if $x^5.y^7={\left(x+y\right)}^{12}$

Find $\frac{dy}{dx}$, if $xy=\log \left(xy\right)$

Find $\frac{dy}{dx}$, if $x^y=y^x$

Find $\frac{dy}{dx}$, if $y{e}^x+x{e}^y=1$

Find $\frac{dy}{dx}$, if $x^2{y}^2={\tan }^{-1}\sqrt{x^2+y^2}+{\cot }^{-1}\sqrt{x^2+y^2}$

Find $\frac{dy}{dx}$, if $\cos \left(xy\right)=x+y$

Differentiate the following function w.r.t.$x$: ${\tan }^{-1}\left(\frac{5-x}{6x^2-5x-3}\right)$.

Find $\frac{dy}{dx}$, if $x^3+x^{2}y+x{y}^2+y^3=81$

Differentiate the following function w.r.t.$x$: ${\tan }^{-1}\left(\frac{a\cos x-b\sin x}{b\cos x+a\sin x}\right)$.

Find $\frac{dy}{dx}$, if $x+\sqrt{xy}+y=1$

Differentiate the following function w.r.t. $x$: ${\tan }^{-1}\left(\frac{7x}{1-12x^2}\right)$.

Differentiate the following function w.r.t. $x$: ${\tan }^{-1}\left(\frac{5ax}{a^2-6x^2}\right)$.

Find $\frac{dy}{dx}$, if $x\sqrt{x}+y\sqrt{y}=a\sqrt{a}$