Partial Derivatives

Anything You Want to Talk About?

A suggestion for problem set 5, the “length of snake” question: Use N or NIntegrate to evaluate the arc length integral numerically. If you ask Mathematica to find that integral symbolically, it will take forever (or at least much longer than you want to wait).

Partial Derivatives

From “Derivatives of a Function of Two Variables” and “Functions of More than Two Variables” in section 3.3 of our textbook.

Key Idea(s)

There is a limit definition that takes limits with respect to a single variable at a time.

To take partial derivatives with respect to x (or any other variable): pretend every variable except x is a constant and use 1-variable derivative rules.

Examples

Find the derivatives with respect to x, y, and z of (x²y + y²x) / z

These derivatives mainly just require treating everything except the variable you’re taking the particular derivative with respect to as if it were a constant:

How about the derivatives with respect to x and y of x tan(x²+y²) - e^xy / 3y

For these derivatives you need to use various differentiation rules from single-variable calculus, in particular the product rule, the quotient rule, and the chain rule. All the single-variable rules can be used if or when needed in taking partial derivatives.

The derivative with respect to x involves the product and chain rules:

The derivative with respect to y involves the chain and quotient rules (or replace the quotient rule with the product rule and a term with a negative exponent):

Partial Derivatives in Mathematica

Use the D function, just as for other derivatives, with the second argument specifying which variable to differentiate with respect to.

For example

D[ x^3 - y^2, x ]

D[ x^3 - y^2, y ]

You can download a notebook with these examples from Canvas.

Theory

Is it possible to have a 2-variable function that, at some point, has a derivative with respect to x but not with respect to y or vice versa? If so, find an example.

We didn’t have time to talk much about this in class, but it’s a worthwhile question to think about if you want.

Problem Set

Problem set 6, on partial derivatives and limits of multivariable functions, is ready.

Work on it after break, and grade it the week of October 17.

See the handout for details.

Next

Second and higher order partial derivatives.

Please read “Higher-Order Partial Derivatives” in section 3.3 of the textbook for Wednesday.

Have a good break in the meantime!