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MathCalculus

Inverse Derivative of 9 − x² on 0 ≤ x ≤ 3 at a = 2

Find f prime at 2, the inverse formula on the stated domain, and its derivative at y=5.

Question

Let f(x)=9−x2f(x)=9-x^2 for 0≤x≤30\le x\le3, with a=2a=2. (a) Find f′(a)f'(a). (b) Express xx as f−1(y)f^{-1}(y). (c) Use (b) to find df−1/dyd f^{-1}/dy at y=f(a)y=f(a).

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Evidence boundary

The inverse is taken only for the original domain 0 ≤ x ≤ 3. The inverse derivative is evaluated at the interior point y = f(2) = 5; its formula applies on 0 < y < 9.