3.5 The Chain Rule

Differentiating Composite Functions like:

y = tan(3x)

These functions require the chain rule:

First, let's find the derivative of a function which we could expand and do traditionally both ways and compare

Traditionally

So =

Using the Chain Rule

The Chain Rule

If g is differentiable at x and f is differentiable at g(x), then the composite function defined by F(x)=f(g(x)) is differentiable and F' is given by the product

F '(x)=f '(g(x))g'(x).

In Leibniz notation, if y = f(u) and u = g(x) are both differentiable functions, then

One version of a Proof:

multiply by and rearrange (if )

= f '(g(x))g'(x)

Practice with finding the "pieces" and differentiating

Power Rule Combined with the Chain Rule

If n is any real number and u = g(x) is differentiable, then

Or Alternately,

Examples:

F(t) = cos(4 - 3t)

Compare and Contrast:

Example of composite, inside of a composite!

y = cos(cos(cos(x)))

Find First and Second Derivative of

#62 If , where

Assignment: 3.5 pg 162; 1-13 odd, 17, 21-25 odd, 29-33 odd, 47-53 odd