Inverse Functions and Inverse Trigonometric Functions Inverse Functions Let π(π₯) be a real-valued function. If π is one-to-one (i.e. passes the horizontal line test), then π has an inverse function (called π β1 (π₯)). The function π and its inverse π have a special relationship, namely that π β1 (π(π₯)) = π₯ and π β1 (π(π₯)) = π₯. Exercise: 1. (a.) Assume that π(π₯) is differentiable at every real π₯ and that π(π₯) is one-to-one. Setting π¦ = π β1 (π₯), find a formula for the derivative of π β1 (π₯). HINT: Use implicit differentiation.
(b.) For which values of π₯ does the derivative of π β1 (π₯) not exist?
2. Suppose that π β² (π₯) =
π₯ 2 +5 7π₯ 2 +17
and that π(1) = 9. Write π = π β1 (π₯). Find πβ² (9).
Inverse Trigonometric Functions 3. Let π(π₯) = arccos π₯. Use implicit differentiation to find π β² (π₯).
4. In your textbook, you may notice that the derivatives of π(π₯) = arcsec π₯ and β(π₯) = 1 β1 arccsc π₯ are given as πβ² (π₯) = and ββ² (π₯) = . Why do the derivatives here 2 2 |π₯|βπ₯ β1
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