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Derivative of an Inverse

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Derivative of an Inverse

- If fis continuous in its domain, then its inverse is continuous on its domain.
- If fis increasing on its domain, then its inverse is increasing on its domain
- If fis decreasing on its domain, then its inverse is decreasing on its domain
- If fis differentiable on an interval containing cand f '(c) does NOT equal 0, then the inverse is differentiable at f (c).

Let’s investigate this…

If f is differentiable at c, the inverse is differentiable at f(c).

f is differentiable at x = 2.

Example:

Since f (2) = 6, g(x) is differentiable at x = 6.

Reciprocals.

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.

Assume that f(x) is differentiable and one-to-one on an interval I with inverse g(x). g(x) is differentiable at any xfor which f '(g(x)) ≠ 0. In particular:

Other Forms:

A function f and its derivative take on the values shown in the table.

If g is the inverse of f, find g'(6).

Let f (x) = x3 + 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 x3+ x – 2 = 0 with a calculator or guess and check)