Partial Derivatives Part 3

Summary

The lesson introduces second-order partial derivatives, which are obtained by differentiating a first partial derivative again.

A mixed (cross) second partial derivative involves differentiating with respect to two different variables, one after the other. In the example, differentiating first with respect to (x) and then with respect to (y) gives (7). Reversing the order—first differentiating with respect to (y), then with respect to (x)—also gives (7).

The notation records the order of differentiation, read from right to left:

• (\frac{\partial^2 f}{\partial y\,\partial x}): differentiate with respect to (x) first, then (y).
• (\frac{\partial^2 f}{\partial x\,\partial y}): differentiate with respect to (y) first, then (x).

In this example, the two mixed partial derivatives are equal. The instructor connects this to Schwarz’s theorem, which states that the mixed second partial derivatives are equal when the relevant continuity condition holds.

Speaker or Sources Featured

• Speaker: One instructor presenting a calculus lesson. No other speakers or sources are identified.