Higher Order Derivatives

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 $\frac{d^{2}y}{d{x}^2}-5\frac{dy}{dx}+6y$

Find third order derivative of $y=e^{\omega x}$, where $\omega$ is a cube root of unity

For $y=\cos \left(m{\sin }^{-1}x\right)$, which of the following is true?

If $f\left(x\right)=10\cos x+\left(13+2x\right)\sin x, \mathrm{ }\mathrm{ }\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}\mathrm{ }{f}^{\prime \prime }\left(x\right)+f\left(x\right)\mathrm{ }$is equal to

Let $f\left(x\right)$ be a function which is differentiable any number of times and $f\left(2x^2-1\right)=2x^{3}f\left(x\right), \forall x\in R$. Then $f^{\left(2010\right)}\left(0\right)=$

$($Here, $f^{\left(n\right)}\left(x\right)=n^{\mathrm{th}}$ order derivative of $f\left(x\right)$$)$

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 $y=e^{3x},$ then $\left(\frac{d^{2}y}{d{x}^2}\right)\left(\frac{d^{2}x}{d{y}^2}\right)$ is

If $y^2=P\left(x\right)$ is a polynomial of degree $3$, then $2\frac{\mathit{d}}{\mathit{d}\mathit{x}}\left(y^3\frac{d^{2}y}{d{x}^2}\right)$ is equal to

Let $P\left(x\right)$ is polynomial of degree four with leading coefficient one and satisfying the condition $P\left(3\right)=11; P\left(-3\right)=11,P^{\text{'}}\left(3\right)=6,P^{\text{'}\text{'}}\left(3\right)=2,$ then value of $P\left(5\right)$ is equal to

If $\sqrt{x+y}+\sqrt{y-x}=a$ then $\frac{d^{2}y}{d{x}^2}$ equals

If $x=e^{\mathit{t}}\sin t$ and $y=e^{\mathit{t}}\cos t$, $\mathrm{t}$ is a parameter, then the value of $\frac{d^{2}x}{d{y}^2}+\frac{d^{2}y}{d{x}^2}$ at $t=0,$ is

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

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