COMPOSITION OF FUNCTIONS Example 1 If and , find . COMPOSITION OF FUNCTIONS

DECOMPOSITION OF FUNCTIONS Example 2 If each function below represents , define and . DECOMPOSITION OF FUNCTIONS

The Chain Rule If y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x, then y = f(g(x)) is a differentiable function of x, and Other ways to write the Rule:

Instructions for The Chain Rule For , to find : Decompose the function Differentiate the MOTHER FUNCTION Differentiate the COMPOSED FUNCTION Multiply the resultant derivatives Substitute for u and Simplify Make sure each function can be differentiated.

Example 1 Find if and . Define f and u: Find the derivative of f and u:

Example 2 Differentiate . Define f and u: Find the derivative of f and u:

Example 3 If f and g are differentiable, , , and ; find . Define h and u: Find the derivative of h and u:

Example 4 Find if . Define f and u: Find the derivative of f and u:

Example 5 Differentiate . Define f and u: Find the derivative of f and u: OR

Now try the Chain Rule in combination with all of our other rules.

Example 1 Differentiate . Use the old derivative rules Chain Rule Twice

Example 2 Find the derivative of the function . Quotient Rule Chain Rule

Example 3 Differentiate . Chain Rule Twice

Example 4 Differentiate . Chain Rule Chain Rule Again

Example 5 Find an equation of the tangent line to at . Find the Derivative Evaluate the Derivative at x = π Find the equation of the line