Question

You can find the derivative of the inverse function by differentiating the inverse explicitly, or by using the formula:

Answer 1

If a function f(x) has an inverse f ⁻¹(x), then by definition

`f\left(f^(-1)(x)\right) = x`

Differentiating both sides with respect to x yields

`f'\left(f^(-1)(x)\right) * \left(f^(-1)\right)'(x) = 1 \implies \left(f^(-1)\right)'(x) = \frac1{f'\left(f^(-1)(x)\right)}`

Now, if f(a) = b and b = f ⁻¹(a), then

`\left(f^(-1)\right)'(a) = \frac1{f'\left(f^(-1)(a)\right)} = \frac1{f'(b)}`