SUNY Geneseo Department of Mathematics

The Chain Rule

Friday, March 24

Math 223
Spring 2023
Prof. Doug Baldwin

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Colloquium

“The Arithmetic Derivative” aka “How to Differentiate a Number?”

Xiao Xiao, Utica College

Monday, March 27, 5:00 - 6:00 PM, Newton 214.

Problem Set

Problem set 8, still on limits and derivatives of multivariable functions, is ready.

Work on it next week, and grade it the week after.

See the handout for more information.

The Chain Rule for Multivariable Functions

Based on “Chain Rules for One or Two Independent Variables” and “The Generalized Chain Rule” in section 3.5 of the textbook.

Key Ideas

The equations for the chain rule with one or two independent variables, and the component derivatives they use (see the book, or examples below, for the actual equations).

The definition of the generalized chain rule, for any number of intermediate and independent variables. See the book for the exact definition.

Remember to rewrite intermediate variables in terms of independent ones after using the chain rule.

The way differentials “cancel” in the chain rule is neat, and provides a nice way to remember it. Here’s what it looks like for single-variable functions, the idea reappears in the chain rule for multivariable functions too.

D X terms seem to cancel in product D F over D X times D X over D T, leaving D F over D T

Examples

Suppose f(x,y) = exy+2y, and x = sin t and y = t2. Use the chain rule to find df/dt.

We started by writing down the appropriate formula, from the chain rule for 1 independent variable:

Functions F of X and Y, X of T, Y of T. D F D T is D X D T times D F D X plus D Y D T times D F D Y

Then we figured out what the derivatives are that it calls for, and plugged them into the formula:

Finding D X D T times D F D X plus D Y D T times D F D Y using a list of derivatives

Finally, we rewrote the remaining occurrences of x and y in terms of t, so there would only be one variable, and the one the derivative implies is the variable of interest, in the answer:

Replacing every X with sine of T and every Y with T squared in a long expression

Finally, we drew a tree diagram to summarize what we did:

Branching diagram showing derivatives for f with respect to x and y, then 1 derivative for each of x and y

If z = (tan x)(cot y), x = s2 + t2, and y = st, find ∂z/∂s and ∂z/∂t.

This time we started with a tree diagram, as a way to work out the derivatives we’d need and how we’d need to combine them:

Branching diagram showing derivatives of z with respect to x and y, then 2 more for each of x and y

Then we pulled out the terms needed for ∂z/∂s and found the derivative:

Multiplying partial derivatives and adding products to find a derivative with the chain rule

Notice that we didn’t have time to write x and y in terms of s and t, although we should do that. Try doing it for yourself over the weekend. You can also try working out what ∂z/∂t is.

Next

Any further discussion you want of the last chain rule example.

Then, partial derivatives tell you the rate at which a function changes if you move in the x direction or the y direction. But what if you move in some intermediate direction (say, along the line y = x)? For that, you want a “directional derivative.”

Please read “Directional Derivatives” in section 3.6 of the textbook.

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