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

$\sqrt{\mathrm{cosec}\left(x^3+1\right)}$

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

$\sin 5x \cos 3x$

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

$\sin 2x\sin x$

$y=\cos 4x \cos 2x$

If $\frac{dy}{dx}=-a \sin 6x-b\sin 2x$, then find the value of $\left(a+b\right)$.

Find $\frac{dy}{dx}$, when:

$y = \sin \left(\frac{1+x^2}{1-x^2}\right)$

Find $\frac{dy}{dx}$, when:

$y=\frac{\left(\sin x+x^2\right)}{\cot 2x}$

If $y=\frac{\left(\cos x-\sin x\right)}{\left(\cos x+\sin x\right)}$, prove that $\frac{\mathit{d}y}{\mathit{d}x}+y^2+1=0$

If $y=\frac{\left(\cos x+\sin x\right)}{\left(\cos x-\sin x\right)}$, prove that $\frac{\mathit{d}y}{\mathit{d}x}={\sec }^2 \left(x+\frac{\pi }{4}\right)$.

Differentiate of the following function w.r.t $x$: $e^{4x}$.