Clicker Question 1 What is the instantaneous rate of change of f (x ) = sin(x) / x when x =  /2 ? A. 2/  B. 0 C. (x cos(x) – sin(x)) / x 2 D. – 4 / 

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Clicker Question 1 What is the instantaneous rate of change of f (x ) = sin(x) / x when x =  /2 ? A. 2/  B. 0 C. (x cos(x) – sin(x)) / x 2 D. – 4 /  2 E. 4 /  2

Clicker Question 2 What is the derivative of g (t ) = e t tan(t ) ? A. e t sec 2 (t ) B. e t (tan(t ) + sec 2 (t )) C. t e t – 1 sec 2 (t ) D. t e t – 1 tan(t ) + e t sec 2 (t ) E. e t sec(t )

The Chain Rule (3/6/09) The Chain Rule tells us how to find the derivative of the composite of two (or more) functions given that we know the individual derivatives. The key idea is that when we compose functions, we multiply their rates of change. This is the most important of the “rules.”

Statement of the Chain Rule If h (x) = f (g (x )), then h ''(x) = f ‘ (g (x )) g ‘ (x) In words, the derivative of a composite function is the derivative of the outer (or last) function with respect to the inner function times the derivative of the inner (or first) function.

Some Examples Use (1) algebra and the Power Rule and (2) the Chain Rule and the Power Rule to find the derivative of f (x) = (x 2 +1) 3 Compare the answers! Find the derivative of f (x) = e 2x three different ways. Compare. In how many ways can you compute the derivative of f (x) = e x^2 ?

Clicker Question 3 According to the Chain Rule, what is the derivative of f (x ) = (4x 2 + 3) 6 ? A. 6 (4x 2 + 3) 5 B. 8x (4x 2 + 3) 5 C. 48x (4x 2 + 3) 5 D. 6 (4x 2 + 3) 5 + 8x E. 6(8x + 3) (4x 2 + 3) 5

Other ways to think about the Chain Rule If you think of the inner function of a composite as a “chunk”  and the outer function as f, then the derivative is f '(  )   ' If g (x) = u and f (u) = y, then dy /dx = dy /du  du /dx You’re not actually canceling fractions here, but it looks that way.

Assignment for Monday (3/16) Read Section 3.4 Do Exercises 7 – 31 odd, 49, 51, 54, 59, and 77. We will a regular class on Monday (not a lab).