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Published byDorothy Greene Modified over 8 years ago
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Derivative of an Inverse
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1980 AB Free Response 3
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Continuity and Differentiability of Inverses 1.If f is continuous in its domain, then its inverse is continuous on its domain. 2.If f is increasing on its domain, then its inverse is increasing on its domain 3.If f is decreasing on its domain, then its inverse is decreasing on its domain 4.If f is differentiable on an interval containing c and f '(c) does NOT equal 0, then the inverse is differentiable at f (c). Let’s investigate this…
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Differentiability of an Inverse f is differentiable at x = 2. Since f (2) = 6, g(x) is differentiable at x = 6. If f is differentiable at c, the inverse is differentiable at f(c). Example: If f '(c) = 0, the inverse is not differentiable at f(c). Example: f '(0) = 0 Since f (0) = 2, g(x) is not differentiable at x = 2. Reciprocals.
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The Derivative of an Inverse Assume that f(x) is differentiable and one- to-one on an interval I with inverse g(x). g(x) is differentiable at any x for which f '(g(x)) ≠ 0. In particular: Other Forms:
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Example 1 A function f and its derivative take on the values shown in the table. xf (x)f '(x) 261/3 683/2 If g is the inverse of f, find g'(6).
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Example 2 Let f (x) = x 3 + x – 2 and let g be the inverse function. Evaluate g'(0). Note: It is difficult to find an equation for the inverse function g. We NEED the formula to evaluate g'(0). (Solve x 3 + x – 2 = 0 with a calculator or guess and check)
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2007 AB Free Response 3
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