1.2.4 A few product and chain rules
I'll wrap up this discussion of velocity vectors with a list of some helpful properties that might look familiar to you from AP Calculus. The proofs of these will show up in the extension questions for this unit.
Suppose are both differentiable paths. Let . Then the following are true:
- If , then
- If is a scalar and , then
- If is a scalar valued function and , then
- If , then
- If , then (A few notes on this one: First of all, note that the path is actually a path in (what we will call a space curve in a few days). Moreover, while usually in a product rule the order in which you differentiate doesn't matter, because cross product is anti-commutative the order here really does matter.)
- If and , then
- If , then