SUNY Geneseo Department of Mathematics
Monday, March 20
Math 223
Spring 2023
Prof. Doug Baldwin
(No.)
Problem set 7, on multivariable functions and their (first) derivatives, is ready now.
Work on it this week, grade it next week.
See the handout for details.
Based on “Higher-Order Partial Derivatives” in section 3.3.
Higher order derivatives are (partial) derivatives of (partial) derivatives.
Clairaut’s Theorem: mixed second (and higher) partial derivatives are equal at any point (a,b) if the mixed derivatives are continuous in some open disk around (a,b).
Does order matter? The only time there’s an issue with the order in which you take derivatives is in mixed derivatives, and then the order almost always doesn’t matter, thanks to Clairaut’s Theorem.
Find all the 2nd partial derivatives of z = x sin y - y cos x. Does Clairaut’s Theorem hold?
A good way to start is to write down the first partial derivatives, then you can find any second derivative by differentiating the appropriate first derivative one more time:
Notice that the last 2 derivatives, the mixed ones, are equal, as Clairaut’s Theorem says they should be.
How about the 2nd partial derivatives of g(s,t) = s ln t?
Follow the same process as above to find the derivative functions:
How if at all does Clairaut’s Theorem describe the mixed derivatives at (0,0)? Notice that the mixed derivative functions are equal, although that function is undefined anywhere t = 0. So you can’t really say the second mixed derivatives are equal at (0,0), and Clairaut’s Theorem doesn’t in this case (because the mixed second derivatives aren’t continuous in any disk around (0,0)).
How about some third derivatives, just to see that you can go farther? They are also derivatives of derivatives found earlier, for example…
Tangent planes and differentiability.
Please read “Tangent Planes,” “Linear Approximations,” and “Differentiability” in section 3.4 of the textbook.