Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 3.10.57

Chapter 3, Problem 3.10.57

Find (f^−1)′(3), where f(x)=x³+x+1.

Verified step by step guidance

Step 1: Understand the problem. We need to find the derivative of the inverse function $ (f^{-1})'(3) $ for the function $ f(x) = x^3 + x + 1 $.

Step 2: Use the formula for the derivative of an inverse function: $ (f^{-1})'(y) = \frac{1}{f'(x)} $ where $ f(x) = y $.

Step 3: Find $ x $ such that $ f(x) = 3 $. This means solving the equation $ x^3 + x + 1 = 3 $ to find the value of $ x $.

Step 4: Compute the derivative $ f'(x) $ of the function $ f(x) = x^3 + x + 1 $. The derivative is $ f'(x) = 3x^2 + 1 $.

Step 5: Evaluate $ f'(x) $ at the $ x $ found in Step 3, and use the formula from Step 2 to find $ (f^{-1})'(3) = \frac{1}{f'(x)} $.

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Key Concepts

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

Inverse Function

An inverse function reverses the effect of the original function. For a function f(x), its inverse f^(-1)(y) satisfies the equation f(f^(-1)(y)) = y. To find the derivative of an inverse function at a specific point, we often use the relationship between the derivatives of the original and inverse functions.

Derivative of Inverse Function

The derivative of an inverse function can be calculated using the formula (f^(-1))'(y) = 1 / f'(x), where y = f(x). This means that to find the derivative of the inverse at a point, we first need to determine the corresponding x-value such that f(x) equals the given y-value, and then compute the derivative of f at that x.

Chain Rule

The chain rule is a fundamental theorem in calculus used to differentiate composite functions. It states that if a function y = f(g(x)) is composed of two functions, the derivative is given by dy/dx = f'(g(x)) * g'(x). This rule is essential when dealing with inverse functions, as it helps in understanding how changes in x affect y through the composition of functions.

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Intro to the Chain Rule

Related Practice

Textbook Question

Watching an elevator An observer is 20 m above the ground floor of a large hotel atrium looking at a glass-enclosed elevator shaft that is 20 m horizontally from the observer (see figure). The angle of elevation of the elevator is the angle that the observer’s line of sight makes with the horizontal (it may be positive or negative). Assuming the elevator rises at a rate of 5 m/s, what is the rate of change of the angle of elevation when the elevator is 10 m above the ground? When the elevator is 40 m above the ground? <IMAGE>

Textbook Question

27–76. Calculate the derivative of the following functions.

$y = \sqrt{10 x + 1}$

Textbook Question

Find the following higher-order derivatives.

d²/dx² (In(x² + 1))

Textbook Question

Suppose f is a one-to-one function with f(2)=8 and f′(2)=4. What is the value of (f^−1)′(8)?

Textbook Question

Use implicit differentiation to find dy/dx.

sin xy = x+y

Textbook Question

Calculate the derivative of the following functions (i) using the fact that b^x = e^{xIn b} and (ii) using logarithmic differentiation. Verify that both answers are the same.

y = (4x+1)^{In x}