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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.10.76
Chapter 3, Problem 3.10.76

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

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First, understand that if y = f(x) is a function, then its inverse function is denoted as x = f^(-1)(y). We need to find the derivative of this inverse function.
Given the function f(x) = x^(2/3), we need to find the inverse function. To do this, set y = x^(2/3) and solve for x in terms of y.
Raise both sides to the power of 3/2 to solve for x: x = y^(3/2). This gives us the inverse function f^(-1)(y) = y^(3/2).
To find the derivative of the inverse function, use the formula: (d/dx)[f^(-1)(y)] = 1 / (f'(f^(-1)(y))).
Calculate f'(x) for the original function: f'(x) = (2/3)x^(-1/3). Substitute f^(-1)(y) = y^(3/2) into this derivative to find the derivative of the inverse function.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Inverse Functions

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.
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Derivative of Inverse Functions

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.
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Power Rule for Derivatives

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.
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Related Practice
Textbook Question

5–8. Calculate dy/dx using implicit differentiation.

sin y+2 = x

Textbook Question

Evaluate the following limits or state that they do not exist. (Hint: Identify each limit as the derivative of a function at a point.)

lim x→π/4 cot x−1 / x−π/4

Textbook Question

Derivatives Find and simplify the derivative of the following functions.

g(t) = t³+3t²+t / t³

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Textbook Question

State the derivative rule for the logarithmic function f(x)=log(subscript b)x. How does it differ from the derivative formula for ln x?

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Textbook Question

Complete the following statement. If dy/dx is small, then small changes in x will result in relatively ______ changes in the value of y.

Textbook Question

Applying the Chain Rule Use the data in Tables 3.4 and 3.5 of Example 4 to estimate the rate of change in pressure with respect to time experienced by the runner when she is at an altitude of 13,330 ft. Make use of a forward difference quotient when estimating the required derivatives.