The Multivariable Chain Rule

Find the indicated derivative. In each case, state the version of the Chain Rule that you are using.

1. $\frac{df}{dt}$, if $f(x,y) = 2x^2y$, $x=\cos(t)$, and $y=\ln(t)$.

2. $\frac{\partial f}{\partial w}$, if $f(x,y) = 2x^2y$, $x=w+z^2$, and $y=\frac{2z+1}{w}$

3. $\frac{\partial f}{\partial v}$, if $f(x,y,z) = 2x^2y+z^3$, $x=u-v+2w$, $y=w2^v-u^3$, and $z = u^2-v$

Let $z = u^2 - v^2$ and suppose that

1. Find the values of $u$ and $v$ that correspond to $x=0$ and $y=2\pi/3$.

2. Use the Chain Rule to find the general partial derivatives and then determine both $\frac{\partial z}{\partial x}\restrict{(0, \frac{2\pi}{3})}{}$ and $\frac{\partial z}{\partial y}\restrict{(0, \frac{2\pi}{3})}{}$

Suppose that the temperature on a metal plate is given by the function $T$ with where the temperature is measured in degrees Fahrenheit and $x$ and $y$ are each measured in feet. Now suppose that an ant is walking on the metal plate in such a way that it walks in a straight line from the point $(1,4)$ to the point $(5,6)$.

1. Find parametric equations $(x(t),y(t))$ for the ant's coordinates as it walks the line from $(1,4)$ to $(5,6)$.

2. What c…

There are several proposed formulas to approximate the surface area of the human body. One model uses the formula where $A$ is the surface area in square meters, $h$ is the height in centimeters, and $w$ is the weight in kilograms.

Since a person's height $h$ and weight $w$ change over time $h$ and $w$ are functions of time $t$. Let us think about what is happening to a child whose height is $60$ centimeters and wei…

Let $z = f(x,y) = 50 - (x+1)^2 - (y+3)^2$ and $z = h(x,y) = 24 - 2x - 6y$.

Suppose a person is walking on the surface $z = f(x,y)$ in such a way that she walks the curve which is the intersection of $f$ and $h$.

1. Show that $x(t) = 4 \cos(t)$ and $y(t) = 4 \sin(t)$ is a parameterization of the “shadow” in the $xy$-plane of the curve that is the intersection of the graphs of $f$ and $h$.

2. Use the parameterization from part (a) to find the instantaneous rate at which her height is changing with respect to time at the ins…