Example 1 If and find COMPOSITION OF FUNCTIONS

Example 1 If and , find COMPOSITION 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 Make sure each function can be differentiated. • Differentiate the MOTHER FUNCTION • Differentiate the COMPOSED FUNCTION • Multiply the resultant derivatives • Substitute for u and Simplify

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

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 Chain Rule . Quotient Rule

Example 3 Differentiate Chain Rule Twice .

Example 4 Differentiate Chain Rule Again

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