The Chain Rule is used for a composition of two functions (in other words, a function within a function) Example: Arnold the paper boy gets paid $25 dollars a week as well as $0.05 per paper delivered. Every week he drops off 20 papers to an apartment building, and then delivers as many as he can, which is 30 papers per hour.

Example continued… Write the number of papers delivered, n, as a function of time, h: n(t) = 30h+20 Write the amount earned, A, as a function of n: A(n)=0.05n + 25 Write the amount earned, A as a function of t. A(t) = 0.05(30h+20) + 25 = 1.5h + 1

Determine the rate of change of amount earned with respect to time A’(t) = 1.5 OR A’(t) = A’[n(t)] x n’(t) = 0.05 x 30 = 1.5 This can also be written as

Example: Find the derivative of OR

In General: Given two differentiable functions g(x) and h(x), the derivative of the composite function f(x) = g[h(x)] is Or if y = f(u) and u = g(x), then

Power of a Function Rule

Examples: Textbook #1d #2c

…Examples continued #3d #5d

…Examples continued #8c #11

…Examples continued Workbook p.39 #16d Homework: Workbook p.118 #1-5,7,8 ace,10,14