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

$f'\left(f^{-1}(x)\right) \times \left(f^{-1}\right)'(x) = 1 \implies \left(f^{-1}\right)'(x) = \dfrac1{f'\left(f^{-1}(x)\right)}$

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

$\left(f^{-1}\right)'(a) = \dfrac1{f'\left(f^{-1}(a)\right)} = \dfrac1{f'(b)}$