More about Partial Derivatives

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Misc

Try out Canvas’s “Blackboard Collaborate” video tool for Wednesday’s class (notice it towards the bottom of Canvas’s navigation bar).

Questions?

Mathematica and Partial Derivatives

Mathematica’s D function does partial derivatives. With multiple variables, find a higher-order derivative with respect to the variables in the order listed. With a list of variables in double brackets, give a vector of derivatives.

See this notebook for examples.

Higher-Order Partial Derivatives

“Higher-Order Partial Derivatives” in section 13.3.

Warning: Our book consistently interprets the “∂-” notation for partial derivatives backwards from how everyone else does it: to everyone else, ∂²f/∂x∂y means ∂/∂x (∂f/∂y).

We will use the conventional notation (given above), not the book’s.

Key Points

Higher order derivatives are possible.

How to evaluate them and interpret the notation.

Derivative with respect to x and then y equals the derivative with respect to y and then x, as long as the second derivatives are continuous (Clairaut’s Theorem).

Example

Find all the second-order derivatives of z = 3x²y² + e^xy.

Start with the 2 first derivatives, then differentiate each of them with respect to each variable:

Does ∂³f/∂x²∂y make sense? Yes, it’s a 3rd derivative, the first derivative taken with respect to y, the second with respect to x, and the third with respect to x again.

Next

Tangent planes, etc.

Read section 13.4.

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