67–78. Derivatives of inverse functions Consider the following functions (on the given interval, if specified). Find the derivative of the inverse function.


f(x) = x^2/3, for x>0

An inverse function reverses the effect of the original function. If f(x) takes an input x and produces an output y, then the inverse function f⁻¹(y) takes y back to x. For a function to have an inverse, it must be one-to-one, meaning it passes the horizontal line test, ensuring each output corresponds to exactly one input.

The derivative of an inverse function can be found using the formula (f⁻¹)'(y) = 1 / f'(x), where y = f(x). This relationship arises from the chain rule, which states that the derivative of the composition of functions is the product of their derivatives. It allows us to compute the slope of the tangent line to the inverse function at a given point.

The power rule is a fundamental technique in calculus for finding the derivative of functions of the form f(x) = x^n, where n is a real number. According to the power rule, the derivative f'(x) = n * x^(n-1). This rule simplifies the process of differentiation, making it easier to compute derivatives of polynomial functions, including those involved in finding derivatives of inverse functions.