Section 2.4 – The Chain Rule

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Presentation transcript:

Section 2.4 – The Chain Rule

Warm-Up Explain why we can not differentiate the function below: Wrong answer: The exponent of 30. Although the exponent makes the derivative difficult, we could use the product rule to find the derivative. Right answer: The function inside of the secant function. We have not taken the derivative of a composition of functions.

Composition of Functions If and , find . COMPOSITION OF FUNCTIONS

Decomposition of Functions 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: Use the Chain Rule: Substitute: Substitute for u: Simplify:

Example 2 Differentiate . Define f and u: Find the derivative of f and u: Use the Chain Rule: Substitute: Substitute for u: Simplify:

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:

White Board Challege Find f '(-2) if:

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

Example 6 Differentiate . Use the old derivative rules Chain Rule Twice

Example 7 Find the derivative of the function . Quotient Rule Chain Rule

Example 8 Differentiate . Product Rule Chain Rule Twice

White Board Challege Find the equation of the tangent to the curve y=3sin(2x) at the point:

Example 9 Differentiate . Chain Rule Chain Rule Again

Example 10 Find an equation of the tangent line to at . Find the Derivative Evaluate the Derivative at x = π Find the equation of the line