A neat little variation on differentiating an integral from last time combines it with the fundamental theorem of calculus. It’s especially interesting in the context of evaluating iterated integrals for irregular regions where the limits of integration may depend on other variables.
Let is a continuous function on the rectangle , and that is also continuous on . Also, let and be two differentiable functions on with images in . Define the function
Then the derivative exists and has the value
In fact, if we forget letting depend on at all, this is the source of some of my favorite questions on first-semester calculus finals.
Anyway, we define another function for the moment
for and in and in . Then .
The fundamental theorem of calculus tells us the first two partial derivatives of immediately, and for the third we can differentiate under the integral sign:
Then we can use the chain rule: