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2 nd Derivatives, and the Chain Rule (2/2/06)

2 nd Derivatives, and the Chain Rule (2/2/06). The second derivative f '' of a function f measures the rate of change of the rate of change of f .

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2 nd Derivatives, and the Chain Rule (2/2/06)

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  1. 2nd Derivatives, and the Chain Rule (2/2/06) • The second derivative f'' of a function f measures the rate of change of the rate of change of f . • On a graph, f'' measures concavity. If f'' is positive, then f is concave up; iff'' is negative, then f is concave down, and if f'' is 0, the curve has no concavity at that point (not bending! E.g., if f is linear). • If the concavity changes from up to down at a point a (so f'' goes from + to -), a is called an inflection point of f.

  2. 2nd Derivative - Units • The units of the second derivative, in general, are output units per input unit per input unit. • Example: f (x) = x 2 + 3x + 2, sof''(x) = 2 . This means that the slope of f everywhere is changing at a rate of 2 vertical units per horizontal unit per horizontal unit. • Example: If position s (in feet) is a function time t (in seconds), then s''(t ) is acceleration, measured in feet/sec/sec.

  3. The Chain Rule • 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.”

  4. Statement of the Chain Rule • If h (x) = f (g (x )), thenh ''(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.

  5. 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) = (x2+1)3Compare the answers! • Find the derivative of f (x) = e2xthree different ways. Compare. • In how many ways can you compute the derivative of f (x) = ex^2

  6. How to think about the Chain Rule, and assignment • If you think of the inner function of a composite as a “chunk”  and the outer function as f , then the derivative isf'()  ' • Assignment for Tuesday: • Read Section 3.5 • Do Exercises 7 – 43 odd, 46, 51, 53, and 65. • On page 240 in Section 3.7, 5 – 13 odd, 43, and 47.

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