Find the derivative of the inverse of the following functions. Express the result with x as the independent variable.
f(x) = x^-1/3

Verified step by step guidance

1. First, understand that you need to find the derivative of the inverse function. If y = f(x), then the inverse function is x = f^(-1)(y).
2. To find the derivative of the inverse function, use the formula: (d/dx)[f^(-1)(x)] = 1 / (f'(f^(-1)(x))).
3. Start by finding the derivative of the original function f(x) = x^(-1/3). Use the power rule: if f(x) = x^n, then f'(x) = n*x^(n-1).
4. Apply the power rule to f(x) = x^(-1/3): f'(x) = (-1/3)*x^(-1/3 - 1) = (-1/3)*x^(-4/3).
5. Now, substitute f'(f^(-1)(x)) into the formula for the derivative of the inverse: (d/dx)[f^(-1)(x)] = 1 / [(-1/3)*(f^(-1)(x))^(-4/3)]. Simplify this expression to express the result with x as the independent variable.

Key Concepts

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In calculus, the derivative is often denoted as f'(x) or df/dx, and it provides critical information about the function's behavior, such as its slope and points of tangency.

Inverse Functions

An inverse function essentially 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. Understanding how to find the derivative of an inverse function is crucial, as it involves applying the relationship between the derivatives of the original and inverse functions.

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y = f(g(x)) is composed of two functions, the derivative can be found by multiplying the derivative of the outer function f with the derivative of the inner function g. This rule is particularly useful when dealing with inverse functions, as it allows for the calculation of derivatives in a structured manner, especially when expressing results in terms of the independent variable.