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Inverse Functions and Inverse Trigonometric Functions Inverse Functions Let 𝑓(𝑥) be a real-valued function. If 𝑓 is one-to-one (i.e. passes the horizontal line test), then 𝑓 has an inverse function (called 𝑓 −1 (𝑥)). The function 𝑓 and its inverse 𝑓 have a special relationship, namely that 𝑓 −1 (𝑓(𝑥)) = 𝑥 and 𝑓 −1 (𝑓(𝑥)) = 𝑥. Exercise: 1. (a.) Assume that 𝑓(𝑥) is differentiable at every real 𝑥 and that 𝑓(𝑥) is one-to-one. Setting 𝑦 = 𝑓 −1 (𝑥), find a formula for the derivative of 𝑓 −1 (𝑥). HINT: Use implicit differentiation.
(b.) For which values of 𝑥 does the derivative of 𝑓 −1 (𝑥) not exist?
2. Suppose that 𝑓 ′ (𝑥) =
𝑥 2 +5 7𝑥 2 +17
and that 𝑓(1) = 9. Write 𝑔 = 𝑓 −1 (𝑥). Find 𝑔′ (9).
Inverse Trigonometric Functions 3. Let 𝑓(𝑥) = arccos 𝑥. Use implicit differentiation to find 𝑓 ′ (𝑥).
4. In your textbook, you may notice that the derivatives of 𝑔(𝑥) = arcsec 𝑥 and ℎ(𝑥) = 1 −1 arccsc 𝑥 are given as 𝑔′ (𝑥) = and ℎ′ (𝑥) = . Why do the derivatives here 2 2 |𝑥|√𝑥 −1
