please help, I don't know what I am doing wrong

Question

Verify that f has an inverse function. Then use the function f and the given real number a to find (f⁻¹)'(a).

f(x) = x³ − 5,    a = 3

Answer 1

To find an inverse function, use "y" for "f(x)":

y = x³ - 5

then swap the "x" and "y" like this:

x = y³ - 5

and then solve for "y":

x + 5 = y³

y = cuberoot(x + 5)

Then replace the "y" with f^-1(x) so:

f^-1(x) = cuberoot(x + 5)

The inverse function derivative rule says:

which, of course, means:

So, I'll pick a = 3

f^-1(3) = cuberoot(a + 5) = cuberoot(8) = 2

Since f(x) = x³ - 5, using the power rule, we can say that f '(x) = 3x² so f '(2) = 3(2²) = 12 so:

[f^-1(3)]' = 1/(f '(f^-1(3)) = 1/f '(2) = 1/12

Answer 2

f(x) = x³ - 5 and f'(x) = 3x²_

Since f is increasing for all x, f is 1-1. So, f has an inverse function.

By inspection, we see that f(2) = 3.

[f^-1(3)]' = 1 / f'(2) = 1/ 12