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. On a graph,
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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. On a graph, f '' measures concavity. If f '' is positive, then f is concave up; if f '' 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 nd 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, so f ''(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.
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.”
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
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 is f '( ) ' 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.