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  $y=\frac{{\sin }^{-1}x}{\sqrt{1-x^2}},$ then $\left(1-x^2\right)\frac{d^{2}y}{d{x}^2}-3x\frac{dy}{dx}-y$ is equal to:

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

If $y=e^{a{\sin }^{-1}x}$, then show that $\left(1-x^2\right)y_{n+2}-\left(2n+1\right)x{y}_{n+1}-\left(n^2+a^2\right)y_n=0$.

If $y={\sin }^{-1}x$, then prove that $\left(1-x^2\right)y_{n+2}-\left(2n+1\right)x{y}_{n+1}-n^2{y}_n=0$

$\mathrm{u}=\log (x^2+y^2)$ then prove $\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }x\mathrm{\partial }y}=\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }y\mathrm{\partial }x}$.

If $f\left(x\right)=e^{x^2}$, then $f^{"}\left(x\right)$ is equal to

Find the $n^{\mathrm{th}}$ derivative of $y=\frac{x-2}{\left(x+2\right)\left(x-3\right)}$

If $f\left(x\right)={\left(x-1\right)}^4{\left(x-2\right)}^3{\left(x-3\right)}^2$, then the value of $f^{\text{'}\text{'}\text{'}}\left(1\right)+f^{"}\left(2\right)+f^{\text{'}}\left(3\right)$ is

If $y=\frac{x}{x^2+a^2},$ then prove that $y_n=\frac{{\left(-1\right)}^{n}n! \cos \left(n+1\right)\theta }{r^{n+1}},$ where $r=\sqrt{x^2+a^2}$ and $\theta ={\tan }^{-1}\left(\frac{a}{x}\right)$

If $y=e^x\ln x$, then find $y_n$.

If $x=a\sec \theta , y=b\tan \theta$, then find $\frac{d^{3}y}{d{x}^3}$ at $x=\frac{\mathrm{\pi }}{6}$.

Find the directional derivative of $\phi = x^{2}yz + 4xz + xyz$ at $(1,2,3)$ in the direction of vector $(2i + j - k).$

$y=e^x\left(a\cos x+b\sin x\right)$, prove that $\frac{d^{2}y}{d{x}^2}-2\frac{dy}{dx}+2y=0$

If $y={\tan }^{-1}x$ then prove that:

${\left(1+x^2\right)}_{y_{n+1}}+2n{x}_{y_n}+n{\left(n-1\right)}_{y_{n-1}}=0$

Find the $n^{\mathrm{th}}$ derivative of $y=\frac{x^3}{x^2-1}$.

If $y\left(x\right)=\frac{1}{1+x}-\frac{x}{{\left(1+x\right)}^2}-\frac{x}{{\left(1+x\right)}^3}-.....-\frac{x}{{\left(1+x\right)}^{20}}$, then $\frac{d^{2}y}{d{x}^2}$ at $x=0$ equals