다변수 연쇄법칙

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

${x}^{2/3}+{y}^{2/3}={a}^{2/3}$

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

$x\,\cos (xy)+y\,\cos \,x=2$

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

${e}^{xy}+y{e}^{y}=1$

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

${x}^{2}{y}^{3}+\cos \,y=0$

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

Find $\frac{dz}{dt}$ using the chain rule where $z=3{x}^{2}{y}^{3},x={t}^{4},$ and $y={t}^{2}.$

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

Let $z=3\,\cos \,x-\sin (xy),x=\frac{1}{t},$ and $y=3t.$ Find $\frac{dz}{dt}.$

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).