The Chain Rule Section 2.4

After this lesson, you should be able to: Find the derivative of a composite function using the Chain Rule. Find the derivative of a function using the General Power Rule. Simplify the derivative using algebra. Find the derivative of a trigonometric function using the Chain Rule.

The Chain Rule (deals with composition of functions) If 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 or,

Extended Power Rule (power rule)     (extended (or general) power rule)

Example Given: y = (6x3 – 4x + 7)3 Then u(x) = 6x3 – 4x + 7 and f(u) = u3 Thus f’(x) = 3(6x3 – 4x + 7)2(18x2 – 4)

Try Which is the, the “inner” function? Which is the , the “outer” function? What is the answer ??

Try Which is the, the “inner” function?= Which is the , the “outer” function? What is the answer ??

Try Which is the, the “inner” function? u = Which is the , the “outer” function? = What is the answer ??

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More Trig Derivatives…

Find the derivative.

Find the derivative.

Book Example Example: Evaluate the derivative of the function at the given point. Use a graphing utility to verify your result. pt: (2, 2)

Example: Given the function and a point: a) Find an equation of the tangent line to the graph of f at a given point. b) Use your calculator to graph the function and its tangent line c) Use the derivative feature on the calculator to confirm your results. pt: (1, 4)