If $x=2\cos \theta -\cos 2\theta$ and $y=2\sin \theta -\sin 2\theta$ find ${\left(\frac{d^{2}y}{d{x}^2}\right)}_{\theta =\frac{\pi }{2}}$.

If $x=a(\theta -\sin \theta ), y=a(1+\cos \theta )$ and $\frac{d^{2}y}{d{x}^2}=\frac{1}{4a}\cos e{c}^m\left(\frac{\theta }{2}\right)$, then the value of $m$ is

Find the second derivative of $5^x$

If the second derivative of $\tan x$ is $k se{c}^{m}x\mathit{ }{\tan }^{n}x,$ then find the value of $k+m+n.$

Find the second derivative of ${\cos }^{-1}x$

If the second derivative of $x\sin x$ is $-x\sin x+k\cos x$, find the value of $k$.

Find the second derivative of $e^{2x}\cos 3x$

If the second derivative of $x^3\log x$ is $x\left(m+n\log x\right)$, then find the value of $m+n$.