## Trending News

Promoted

# AP Calculus Derivatives of Inverse Functions?

Verify that f has an inverse. Then use the function f and the given real number a to find

(f ^−1)'(a).

f(x) = 1/27(x^5 + 2x^3) a = −11

Find (f^-1)'(-11)

### 1 Answer

Relevance

- kbLv 73 years ago
Observe that f'(x) = (1/27)(5x^4 + 6x^2) is non-decreasing. Hence, f has an inverse.

When a (= y) = -11, we find that x = -3, because f(-3) = -11.

So, the Inverse Function Theorem yields

(f^(-1))'(-11) = 1/f'(x) {at x = -3}

....................= 1/[(1/27)(5x^4 + 6x^2)] {at x = -3}

....................= 27/(x^2 * (5x^2 + 6)) {at x = -3}

....................= 27/(9 * (45 + 6))

....................= 1/17.

I hope this helps!

- Login to reply the answers

Still have questions? Get your answers by asking now.