Chain rule for higher-order partial derivatives and differentials of a composite function

Prove: if

then

where $t=xyz$, and find the function $F$.

Find the differential of the indicated order: $d^{n}u$, if $u=f(x+y+z)$.

Find the first- and second-order derivatives of the following composite function:$u=f(x^{2}+y^{2}+z^{2})$ .

Find the first- and second-order derivatives of the following composite function:$u=f\left(x, \dfrac {x}{y}\right)$ .

Find the first- and second-order derivatives of the following composite function:$u=f(x, xy, xyz)$ .

Let $u=f(x+y,xy)$, find $\dfrac {\partial^{2} u}{\partial x \partial y}$.

Let $u=f(x+y+z, x^{2}+y^{2}+z^{2})$, find $\Delta u=\dfrac {\partial^{2} u}{\partial x^{2}}+\dfrac {\partial^{2} u}{\partial y^{2}}+\dfrac {\partial^{2} u}{\partial z^{2}}$.

Find the first- and second-order total differentials of the following composite function ($x, y$ and $z$ are the independent variables):$u=f(t)$, where $t=x+y$.

Find the first- and second-order total differentials of the following composite function ($x, y$ and $z$ are the independent variables):$u=f(t)$, where $t=\dfrac {y}{x}$.

Find the first- and second-order total differentials of the following composite function ($x, y$ and $z$ are the independent variables):$u=f(\sqrt{x^{2}+y^{2}})$ .

Find the first- and second-order total differentials of the following composite function ($x, y$ and $z$ are the independent variables):$u=f(t)$, where $t=xyz$.

Find the first- and second-order total differentials of the following composite function ($x, y$ and $z$ are the independent variables):$u=f(x^{2}+y^{2}+z^{2})$ .

Find the first- and second-order total differentials of the following composite function ($x, y$ and $z$ are the independent variables):$u=f(\xi ,\eta )$, where $\xi =ax$, $\eta =by$.

Find the first- and second-order total differentials of the following composite function ($x, y$ and $z$ are the independent variables):$u=f(\xi ,\eta )$, where $\xi =x+y$, $\eta =x-y$.

Find the first- and second-order total differentials of the following composite function ($x, y$ and $z$ are the independent variables):$u=f(\xi ,\eta )$, where $\xi =xy$, $\eta =\dfrac {x}{y}$.

Find the first- and second-order total differentials of the following composite function ($x, y$ and $z$ are the independent variables):$u=f(x+y,z)$ .

Find the first- and second-order total differentials of the following composite function ($x, y$ and $z$ are the independent variables):$u=f(x+y+z, x^{2}+y^{2}+z^{2})$ .

Find the first- and second-order total differentials of the following composite function ($x, y$ and $z$ are the independent variables):$u=f\left(\dfrac {x}{y}, \dfrac {y}{z}\right)$ .

Find the first- and second-order total differentials of the following composite function ($x, y$ and $z$ are the independent variables):$u=f(x,y,z)$, where $x=t$, $y=t^{2}$, $z=t^{3}$.

Find the first- and second-order total differentials of the following composite function ($x, y$ and $z$ are the independent variables):$u=f(\xi ,\eta ,\zeta )$, where $\xi =ax$, $\eta =by$, $\zeta =cz$.

Find the first- and second-order total differentials of the following composite function ($x, y$ and $z$ are the independent variables):$u=f(\xi ,\eta ,\zeta )$, where $\xi =x^{2}+y^{2}$, $\eta =x^{2}-y^{2}$, $\zeta =2xy$.

Find $d^{n}u$, given:$u=f(ax+by+cz)$ .

Find $d^{n}u$, given:$u=f(ax,by,cz)$ .

Find $d^{n}u$, given:$u=f(\xi ,\eta ,\zeta )$, where $\xi =a_{1}x+b_{1}y+c_{1}z$, $\eta =a_{2}x+b_{2}y+c_{2}z$, $\zeta =a_{3}x+b_{3}y+c_{3}z$.

Let $u=f(r)$, where $r=\sqrt{x^{2}+y^{2}+z^{2}}$ and $f$ is a twice-differentiable function. Prove that:

where $\Delta u=\dfrac {\partial^{2} u}{\partial x^{2}}+\dfrac {\partial^{2} u}{\partial y^{2}}+\dfrac {\partial^{2} u}{\partial z^{2}}$, $\Delta$ is the Laplace operator, and find the function $F$.

Prove that if the function $u=u(x,t)$ satisfies the heat conduction equation (see problem 3309: $\dfrac {\partial u}{\partial t}=a^{2}\dfrac {\partial^{2} u}{\partial x^{2}}$), then the function

also satisfies this equation.

If $z=\sin y+f(\sin x-\sin y)$, where $f$ is a differentiable function. Simplify the expression

If

where $f$ is a differentiable function. Simplify the expression

Let:

and

Prove that:

Let $u=u(x,y)$ be a differentiable function such that when $y=x^{2}$ we have

and

Find $\dfrac {\partial u}{\partial y}$ when $y=x^{2}$.

For the function $f(u)$ find $f_{x}^{\prime }$ and $f_{y}^{\prime }$, if:

For the function $f(u ; v)$ find $f_{x}^{\prime }$ and $f_{y}^{\prime }$, if:

Find $f_{y}^{\prime }\left(x ; x^{2}\right)$ for a differentiable function $f(x ; y)$ satisfying the conditions