다변수 미분 전체 문제

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

Let $z={e}^{1-xy},x={t}^{1/3},$ and $y={t}^{3}.$ Find $\frac{dz}{dt}.$

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

Find $\frac{dz}{dt}$ by the chain rule where $z={\cosh }^{2}(xy),x=\frac{1}{2}t,$ and $y={e}^{t}.$

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

Let $z=\frac{x}{y},x=2\,\cos \,u,$ and $y=3\,\sin \,v.$ Find $\frac{\partial z}{\partial u}$ and $\frac{\partial z}{\partial v}.$

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 $\frac{\partial w}{\partial t}.$

For the following exercises, use this information: A function $f(x,y)$ is said to be homogeneous of degree $n$ if $f(tx,ty)={t}^{n}f(x,y).$ For all homogeneous functions of degree $n,$ the following equation is true: $x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}=nf(x,y).$ Show that the given function is homogeneous and verify that $x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}=nf(x,y).$

$f(x,y)=3{x}^{2}+{y}^{2}$

For the following exercises, use this information: A function $f(x,y)$ is said to be homogeneous of degree $n$ if $f(tx,ty)={t}^{n}f(x,y).$ For all homogeneous functions of degree $n,$ the following equation is true: $x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}=nf(x,y).$ Show that the given function is homogeneous and verify that $x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}=nf(x,y).$

$f(x,y)=\sqrt{{x}^{2}+{y}^{2}}$

For the following exercises, use this information: A function $f(x,y)$ is said to be homogeneous of degree $n$ if $f(tx,ty)={t}^{n}f(x,y).$ For all homogeneous functions of degree $n,$ the following equation is true: $x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}=nf(x,y).$ Show that the given function is homogeneous and verify that $x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}=nf(x,y).$

$f(x,y)={x}^{2}y-2{y}^{3}$

The volume of a right circular cylinder is given by $V(x,y)=\pi {x}^{2}y,$ where $x$ is the radius of the cylinder and y is the cylinder height. Suppose $x$ and $y$ are functions of $t$ given by $x=\frac{1}{2}t$ and $y=\frac{1}{3}t$ so that $x\,\text{and}\,y$ are both increasing with time. How fast is the volume increasing when $x=2$ and $y=\frac{4}{3}?$

The pressure $P$ of a gas is related to the volume and temperature by the formula $PV=kT,$ where temperature is expressed in kelvins. Express the pressure of the gas as a function of both $V$ and $T.$ Find $\frac{dP}{dt}$ when $k=1,$ $\frac{dV}{dt}=2$ cm³/min, $\frac{dT}{dt}=\frac{1}{2}$ K/min, $V=20$ cm³, and $T=20^\circ \text{F}.$

The radius of a right circular cone is increasing at $3$ cm/min whereas the height of the cone is decreasing at $2$ cm/min. Find the rate of change of the volume of the cone when the radius is $13$ cm and the height is $18$ cm.

The volume of a frustum of a cone is given by the formula $V=\frac{1}{3}\pi z({x}^{2}+{y}^{2}+xy),$ where $x$ is the radius of the smaller circle, $y$ is the radius of the larger circle, and $z$ is the height of the frustum (see figure). Find the rate of change of the volume of this frustum when $x=10\,\text{in}.,\,y=12\,\text{in.,}\,\text{and}\,z=18\,\text{in}.$ if $\frac{dz}{dt}=-5,\frac{dx}{dt}=1,\frac{dy}{dt}=1$ (all in/min).

A closed box is in the shape of a rectangular solid with dimensions $x,y,\,\text{and}\,z.$ (Dimensions are in inches.) Suppose each dimension is changing at the rate of $0.5$ in./min. Find the rate of change of the total surface area of the box when $x=2\,\text{in}.,\,y=3\,\text{in.,}\,\text{and}\,z=1\,\text{in}.$

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.

The temperature $T$ at a point $(x,y)$ is $T(x,y)$ and is measured using the Celsius scale. A fly crawls so that its position after $t$ seconds is given by $x=\sqrt{1+t}$ and $y=2+\frac{1}{3}t,$ where $x\,\text{and}\,y$ are measured in centimeters. The temperature function satisfies ${T}_{x}(2,3)=4$ and ${T}_{y}(2,3)=3.$ How fast is the temperature increasing on the fly’s path after $3$ sec?

The $x\,\text{and}\,y$ components of a fluid moving in two dimensions are given by the following functions: $u(x,y)=2y$ and $v(x,y)=-2x;$ $x\ge 0;y\ge 0.$ The speed of the fluid at the point $(x,y)$ is $s(x,y)=\sqrt{u{(x,y)}^{2}+v{(x,y)}^{2}}.$ Find $\frac{\partial s}{\partial x}$ and $\frac{\partial s}{\partial y}$ using the chain rule.

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 $\text{u}=\left(\cos \,\frac{\pi }{4}\right)i+\left(\sin \,\frac{\pi }{4}\right)j$