Textbook Question
47–56. Derivatives of inverse functions at a point Consider the following functions. In each case, without finding the inverse, evaluate the derivative of the inverse at the given point.
f(x)=4e^10x; (4,0)
Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 3.10.71
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) = e^3x+1
Verified step by step guidance1
First, understand that if y = f(x) = e^(3x+1), then the inverse function, denoted as f^(-1)(y), is the function that satisfies x = f^(-1)(y).
To find the derivative of the inverse function, we use the formula: (f^(-1))'(y) = 1 / f'(x), where x = f^(-1)(y).
Calculate the derivative of the original function f(x) = e^(3x+1). Using the chain rule, the derivative f'(x) = 3e^(3x+1).
Substitute f'(x) into the inverse derivative formula: (f^(-1))'(y) = 1 / (3e^(3x+1)).
Since y = e^(3x+1), solve for x in terms of y to express x = f^(-1)(y). Then substitute this expression for x back into the formula for (f^(-1))'(y) to find the derivative of the inverse function in terms of y.
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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 essentially reverses the effect of the original function. If a function f takes an input x and produces an output y, the inverse function f⁻¹ takes y back to x. Understanding how to find and work with inverse functions is crucial for solving problems involving derivatives of these functions.
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Inverse Cosine
Derivative of a Function
The derivative of a function measures how the function's output changes as its input changes. It is a fundamental concept in calculus that provides information about the slope of the function at any given point. For the function f(x) = e^(3x+1), finding its derivative is the first step in determining the derivative of its inverse.
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Derivative of Inverse Functions Formula
The derivative of an inverse function can be found using the formula (f⁻¹)'(y) = 1 / f'(x), where y = f(x). This relationship shows that the derivative of the inverse function at a point is the reciprocal of the derivative of the original function at the corresponding point. This concept is essential for calculating the derivative of the inverse function in the given problem.
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Related Practice
Textbook Question
Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
f(v) = v¹⁰⁰+e^v+10
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Textbook Question
Derivative calculations Evaluate the derivative of the following functions at the given point.
f(s) = 2√s-1; a=25
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Textbook Question
27–40. Implicit differentiation Use implicit differentiation to find dy/dx.
6x³+7y³ = 13xy
Textbook Question
Simplify the expression e^xln(x²+1).
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Textbook Question
Find the slope of the line tangent to the graph of f(x) = x / x+6 at the point (3, 1/3) and at (-2, -1/2).