Second Order Derivative

The function $f:R\to R$ satisfies $f\left(x^2\right)·f^{"}\left(x\right)=f^{\text{'}}\left(x\right)·f^{\text{'}}\left(x^2\right)$ for all real $x$ $.$ Given that $f\left(1\right)=1$ and $f^{\text{'}\text{'}\text{'}}\left(1\right)=8,$ compute the value of $f^{\text{'}}\left(1\right)+f^{"}\left(1\right).$

If $2x=y^{\frac{1}{5}}+\frac{1}{y^{\frac{1}{5}}}$ then $\left(x^2-1\right)\frac{{\text{d}}^{2}y}{\text{d}{x}^2}+x\frac{\text{d}y}{\text{d}x}=k\mathrm{y,}$then find the value of $k$

If $y=e^x\left(\sin x+\cos x\right)$, then what would be the value of  $\frac{d^{2}y}{d{x}^2}-2\frac{dy}{dx}+2y$ :

If $y=3e^{2x}+2e^{3x}$, then what would be the value of given expression $\frac{d^{2}y}{d{x}^2}-5\frac{dy}{dx}+6y$

Let $f\left(x\right)=x\mathrm{t}\mathrm{a}{\mathrm{n}}^{-1}\left(x^2\right)+x^4$. Let $f^k\left(x\right)$ denote the $k^{\mathrm{th}}$ derivative of $f\left(x\right)$ w.r.t. $x$, $k\in N$. If $f^{2m}\left(0\right)\ne 0, m\in N,$ then $m$ equals to.....

If $y=x-e^{2x}$, then $\frac{d^{2}x}{d{y}^2}$ is equal to

If $y=x\cos x$, then $y^{"}\left(\frac{\mathrm{\pi }}{2}\right)+y\left(\frac{\mathrm{\pi }}{2}\right)=$

If $y=x\log x,$ then $\frac{d^{2}y}{ d{x}^2}$ is

If $y={\cos }^{-1}x$, find $\frac{d^{2}y}{d{x}^2}$ in terms of $y$ alone.

If $y={\left({\tan }^{-1}x\right)}^2$, show that 

${\left(x^2+1\right)}^2{y}_2+2x\left(x^2+1\right)y_1=2$

If $e^y\left(x+1\right)=1$ show that $y_2={y_1}^2$

If $y=5\cos x-3\sin x$, prove that $\frac{d^{2}y}{d{x}^2}+y=0$.

If $y=\sin \left(m{\sin }^{-1}x\right)$, show that $\left(1-x^2\right)\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}+m^{2}y=0$.

If $y=e^{m{\cos }^{-1}x}$, show that

$\left(1-x^2\right)\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}-m^{2}y=0$

If $y=3\cos \left(\log x\right)+4\sin \left(\log x\right)$, show that $x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}+y=0$.

If $y=\log \left(\log 2x\right)$, show that $x\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}+x{\left(\frac{dy}{dx}\right)}^2=0$.

If $y={\left(x+\sqrt{x^2-1}\right)}^m$, show that  $\left(x^2-1\right)\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}=m^{2}y$.

If $x^m{y}^n={\left(x+y\right)}^{m+n}$, show that $\frac{d^{2}y}{d{x}^2}=0$

If $y={\sin }^{-1}x$, show that

$\left(1-x^2\right)\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}=0$