4.5 The chain rule  (Page 6/9)

$f\left(x,y\right)=x^2+y^2,$ $x=t,y=t^2$

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

$f\left(x,y\right)=xy,x=1-\sqrt{t},y=1+\sqrt{t}$

$f\left(x,y\right)=\frac{x}{y},x=e^t,y=2e^t$

$f\left(x,y\right)=\text{ln}\left(x+y\right),$ $x=e^t,y=e^t$

$f\left(x,y\right)=x^4,$ $x=t,y=t$

Let $w\left(x,y,z\right)=x^2+y^2+z^2,$ $x=\text{cos}t,y=\text{sin}t,$ and $z=e^t.$ Express $w$ as a function of $t$ and find $\frac{dw}{dt}$ directly. Then, find $\frac{dw}{dt}$ using the chain rule.

Let $z=x^{2}y,$ where $x=t^2$ and $y=t^3.$ Find $\frac{dz}{dt}.$

Let $u=e^x\text{sin}y,$ where $x=t^2$ and $y=\pi t.$ Find $\frac{du}{dt}$ when $x=\text{ln}2$ and $y=\frac{\pi }{4}.$

$\text{sin}\left(6x\right)+\text{tan}\left(8y\right)+5=0$

$x^3+y^{2}x-3=0$

$\text{sin}\left(x+y\right)+\text{cos}\left(x-y\right)=4$

$x^2-2xy+y^4=4$

$x{e}^y+y{e}^x-2x^{2}y=0$

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

$x\text{cos}(xy)+y\text{cos}x=2$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 $y=5?$

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 $k=1,$ $\frac{dV}{dt}=2$ cm ³ /min, $\frac{dT}{dt}=\frac{1}{2}$ K/min, $V=20$ cm ³ , and $T=20\text{°}\text{F}.$

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 $V=\frac{1}{3}\pi z\left(x^2+y^2+xy\right),$ 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 $x=10\text{in}\text{.,}y=12\text{in.,}\text{and}z=18\text{in}.$

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 $x=2\text{in}\text{.,}y=3\text{in.,}\text{and}z=1\text{in}.$

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 $100\text{Ω},$ the y resistance is $200\text{Ω},$ and the $z$ resistance is $300\text{Ω}.$ Also, suppose the $x$ resistance is changing at a rate of $2\text{Ω}\text{/}\text{min},$ the $y$ resistance is changing at the rate of $1\text{Ω}\text{/}\text{min},$ 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 $x\text{and}t$ are measured in centimeters. The temperature function satisfies $T_x\left(2,3\right)=4$ and $T_y\left(2,3\right)=3.$ 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 $v(x,y)=\mathrm{-2}x;$ $x\ge 0;y\ge 0.$ 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 $u=u\left(x,y,z\right),$ 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 $\frac{\partial u}{\partial r}.$