The Chain Rule

Composite Functions When a function is composed of an inner function and an outer function, it is called a “composite function” When a relationship can be described using 2 functions.

The elevation (m) of an airplane is a function of time (s) The % of 0 2 in the atmosphere is a function of elevation t e %

t e % We could then say the % of O 2 is a function of time… The polynomial equation that related these variables would be a “composite” function

WhalesWhales example

To differentiate a composite function, we use The Chain Rule

Differentiate f(x) = x 3 f’(x) = 3x 2 What if “x” was actually not just a variable, but another function altogether

Differentiate f(x) = (2x + 1) 3 The chain rule states 1. Differentiate the outer function first f’(x) = 3(2x + 1) 2 2. Differentiate the inner function second f’(x) = 3(2x + 1) 2 (2) f’(x) = 6(2x + 1) 2 3. Simplify

The Chain Rule Given two differentiable functions g(x) and h(x), the derivative of the composite function f(x) = g[h(x)] is f’(x) = g’[h(x)] X h’(x)

Determine the equation of the tangent to f(x) = 3x(1-x) 2 y = -0.75x