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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 3.10.54
Chapter 3, Problem 3.10.54

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)

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Step 1: Understand the problem. We need to find the derivative of the inverse function of f(x) = 4e^{10x} at the point (4, 0). This means we need to find (f^{-1})'(0).
Step 2: Use the formula for the derivative of an inverse function. If y = f(x) and f is invertible, then the derivative of the inverse function at a point is given by (f^{-1})'(y) = 1 / f'(x), where f(x) = y.
Step 3: Identify the point of interest. We are given the point (4, 0), which means f(x) = 4 when x = 0. Therefore, we need to find f'(0).
Step 4: Differentiate the function f(x) = 4e^{10x}. The derivative f'(x) is found using the chain rule: f'(x) = 4 * 10 * e^{10x} = 40e^{10x}.
Step 5: Evaluate the derivative at x = 0. Substitute x = 0 into f'(x) to find f'(0) = 40e^{0} = 40. Now, use the inverse derivative formula: (f^{-1})'(0) = 1 / f'(0) = 1 / 40.

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Key Concepts

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

Inverse Functions

Inverse functions are functions that 'reverse' 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. Understanding how to find and work with inverse functions is crucial for evaluating derivatives of inverses.
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Inverse Cosine

Derivative of Inverse Functions

The derivative of an inverse function can be calculated using the formula (f⁻¹)'(y) = 1 / f'(x), where y = f(x). This relationship shows how the rate of change of the inverse function at a point is related to the rate of change of the original function at the corresponding point. This concept is essential for solving problems involving derivatives of inverse functions.
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Derivatives of Inverse Sine & Inverse Cosine

Exponential Functions and Their Derivatives

Exponential functions, such as f(x) = 4e^(10x), have specific properties and derivatives. The derivative of an exponential function is proportional to the function itself, specifically f'(x) = k * e^(kx) for some constant k. Understanding how to differentiate exponential functions is necessary for applying the derivative of the inverse function formula.
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Derivatives of General Exponential Functions
Related Practice
Textbook Question

A surface ship is moving (horizontally) in a straight line at 10 km/hr. At the same time, an enemy submarine maintains a position directly below the ship while diving at an angle that is 20° below the horizontal. How fast is the submarine’s altitude decreasing?

Textbook Question

15–48. Derivatives Find the derivative of the following functions.

y = In (x³+1)^π

Textbook Question

Find the derivative of the following functions.

y = x sin x

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

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

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).