13.5 Day 1 Chain Rule for Functions of Several Variables From Calc AB Chain Rule: dy dy du dx du dx If f g is the composite of y f u and u g x , then: f g fat u g x gat x example: f x sin x g x x2 4 f x cos x g x 2x Find: f g at x 2 g 2 4 4 0 f 0 g 2 cos 0 2 2 1 4 4 We could also do it this way: f g x sin x 2 4 Evaluate at x=2 y sin x 2 4 dy cos x 2 4 2 x dx dy cos 22 4 2 2 dx dy cos 0 4 dx dy 4 dx Example 1 How do you think we should do this problem? Solve via substitution Now we can do example 1 without first doing a substitution Solution to Example 1 Example 2 Two objects are traveling in elliptical paths given by the parametric equations given below. What is the rate of change of the distance between the objects when t = π? Hint use the Chain rule: Example 2 Solution Solution: You can see that the distance s between the two objects is given by Example 2 solution continued Example 3 Find the answer by using substitution Example 3 solution 1 We can now do example 3 without first having to do a substitution There are some things that you Just need to memorize!!!