Derivative of an Inverse Function

Compute $f'(a)$ directly, take its reciprocal, substitute back, and simplify

Identify the y-value corresponding to $a$, differentiate $f^{-1}(y)$ with respect to $y$, and then substitute $y=a$

Differentiate the inverse function directly using the chain rule, then compute $f'(a)$, and take the reciprocal

Identify the x-value where $f(x)=a$, compute $f'(x)$ at that point, take the reciprocal to obtain $\left(f^{-1}\right)'(a)$, and simplify the result