다변수 미분 전체 문제

For the following exercises, find $\frac{dy}{dx}$ using partial derivatives.

Let $z={e}^{{x}^{2}y},$ where $x=\sqrt{uv}$ and $y=\frac{1}{v}.$ Find $\frac{\partial z}{\partial u}$ and $\frac{\partial z}{\partial v}.$

For the following exercises, find $\frac{dy}{dx}$ using partial derivatives.

If $z=xy{e}^{x/y},$ $x=r\,\cos \,\theta ,$ and $y=r\,\sin \,\theta ,$ find $\frac{\partial z}{\partial r}$ and $\frac{\partial z}{\partial \theta }$ when $r=2$ and $\theta =\frac{\pi }{6}.$

For the following exercises, find $\frac{dy}{dx}$ using partial derivatives.

Find $\frac{\partial w}{\partial s}$ if $w=4x+{y}^{2}+{z}^{3},x={e}^{r{s}^{2}},y=\ln \left(\frac{r+s}{t}\right),$ and $z=rs{t}^{2}.$

For the following exercises, find $\frac{dy}{dx}$ using partial derivatives.

If $w=\sin (xyz),x=1-3t,y={e}^{1-t},$ and $z=4t,$ find $\frac{\partial w}{\partial t}.$

The total resistance in a circuit that has three individual resistances represented by $x,y,$ and $z$ is given by the formula $R(x,y,z)=\frac{xyz}{yz+xz+xy}.$ Suppose at a given time the $x$ resistance is $100\Omega ,$ the y resistance is $200\Omega ,$ and the $z$ resistance is $300\Omega .$ Also, suppose the $x$ resistance is changing at a rate of $2\Omega /\min ,$ the $y$ resistance is changing at the rate of $1\Omega /\min ,$ and the $z$ resistance has no change. Find the rate of change of the total resistance in this circuit at this time.

Let $u=u(x,y,z),$ where $x=x(w,t),y=y(w,t),z=z(w,t),w=w(r,s),\text{and}\,t=t(r,s).$ Use a tree diagram and the chain rule to find an expression for $\frac{\partial u}{\partial r}.$

For the following exercises, find the directional derivative using the limit definition only.

$f(x,y)=5-2{x}^{2}-\frac{1}{2}{y}^{2}$ at point $P(3,4)$ in the direction of $\text{u}=\left(\cos \,\frac{\pi }{4}\right)i+\left(\sin \,\frac{\pi }{4}\right)j$

For the following exercises, find the directional derivative using the limit definition only.

$f(x,y)={y}^{2}\cos (2x)$ at point $P\left(\frac{\pi }{3},2\right)$ in the direction of $\text{u}=\left(\cos \,\frac{\pi }{4}\right)i+\left(\sin \,\frac{\pi }{4}\right)j$

Find the directional derivative of $f(x,y)={y}^{2}\sin (2x)$ at point $P\left(\frac{\pi }{4},2\right)$ in the direction of $u=5i+12j.$

For the following exercises, find the directional derivative of the function at point $P$ in the direction of $u$ or $v$ as appropriate.

$f(x,y)=xy,$ $P(0,-2),$ $v=\frac{1}{2}i+\frac{\sqrt{3}}{2}j$

For the following exercises, find the directional derivative of the function at point $P$ in the direction of $u$ or $v$ as appropriate.

$h(x,y)={e}^{x}\sin \,y,P\left(1,\frac{\pi }{2}\right),v=-i$

For the following exercises, find the directional derivative of the function at point $P$ in the direction of $u$ or $v$ as appropriate.

$h(x,y,z)=xyz,P(2,1,1),v=2i+j-k$

For the following exercises, find the directional derivative of the function at point $P$ in the direction of $u$ or $v$ as appropriate.

$f(x,y)=xy,P(1,1),u=\langle \frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\rangle$

For the following exercises, find the directional derivative of the function at point $P$ in the direction of $u$ or $v$ as appropriate.

$f(x,y)={x}^{2}-{y}^{2},\begin{array}{cc}u=\langle \frac{\sqrt{3}}{2},\frac{1}{2}\rangle , & P(1,0)\end{array}$

For the following exercises, find the directional derivative of the function at point $P$ in the direction of $u$ or $v$ as appropriate.

$f(x,y)=3x+4y+7,\begin{array}{cc}u=\langle \frac{3}{5},\frac{4}{5}\rangle , & P\left(0,\frac{\pi }{2}\right)\end{array}$

For the following exercises, find the directional derivative of the function at point $P$ in the direction of $u$ or $v$ as appropriate.

$\begin{array}{ccc}f(x,y)={e}^{x}\cos \,y, & u=\langle 0,1\rangle , & P=\left(0,\frac{\pi }{2}\right)\end{array}$

For the following exercises, find the directional derivative of the function at point $P$ in the direction of $u$ or $v$ as appropriate.

$\begin{array}{ccc}f(x,y)={y}^{10}, & u=\langle 0,-1\rangle , & P=(1,-1)\end{array}$

For the following exercises, find the directional derivative of the function at point $P$ in the direction of $u$ or $v$ as appropriate.

$f(x,y)=\ln ({x}^{2}+{y}^{2}),\begin{array}{cc}u=\langle \frac{3}{5},\frac{4}{5}\rangle , & P(1,2)\end{array}$

For the following exercises, find the directional derivative of the function at point $P$ in the direction of $u$ or $v$ as appropriate.

$f(x,y)={x}^{2}y,\begin{array}{cc}P(-5,5), & v=3i-4j\end{array}$

For the following exercises, find the directional derivative of the function at point $P$ in the direction of $u$ or $v$ as appropriate.

$f(x,y,z)={y}^{2}+xz,\begin{array}{cc}P(1,2,2), & v=\langle 2,-1,2\rangle \end{array}$