Skip to main content
Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.9.19
Chapter 4, Problem 4.9.19

Finding antiderivatives. Find all the antiderivatives of the following functions. Check your work by taking derivatives.


ƒ(x) = eˣ

Verified step by step guidance
1
Step 1: Recall the definition of an antiderivative. An antiderivative of a function ƒ(x) is a function F(x) such that F'(x) = ƒ(x).
Step 2: Identify the given function ƒ(x) = eˣ. Note that the derivative of eˣ is itself, meaning eˣ is its own antiderivative.
Step 3: Write the general form of the antiderivative. Since the derivative of a constant is zero, the antiderivative of eˣ includes an arbitrary constant C. Thus, F(x) = eˣ + C.
Step 4: Verify your result by differentiating F(x). Compute F'(x) = d/dx [eˣ + C]. Using the derivative rules, F'(x) = eˣ, which matches the original function ƒ(x).
Step 5: Conclude that all antiderivatives of ƒ(x) = eˣ are given by F(x) = eˣ + C, where C is an arbitrary constant.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

Key Concepts

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

Antiderivative

An antiderivative of a function is another function whose derivative is the original function. In calculus, finding antiderivatives is essential for solving problems related to integration. The general form of an antiderivative includes a constant of integration, C, since the derivative of a constant is zero, leading to multiple valid antiderivatives.
Recommended video:
05:50
Antiderivatives

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a^x, where 'a' is a positive constant. The function eˣ, where e is Euler's number (approximately 2.718), is a special case of an exponential function that has unique properties, particularly that its derivative is equal to itself. This property simplifies the process of finding antiderivatives for functions involving e.
Recommended video:
6:13
Exponential Functions

Verification by Differentiation

Verification by differentiation involves taking the derivative of the antiderivative found to ensure it matches the original function. This step is crucial in confirming the correctness of the antiderivative. If the derivative of the antiderivative equals the original function, it validates the solution and demonstrates a proper understanding of the relationship between differentiation and integration.
Recommended video:
05:53
Finding Differentials
Related Practice
Textbook Question

Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.


ƒ(x) = sin 3x on [-π/4,π/3]

Textbook Question

Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.


ƒ(x) = x² - 10 on [-2, 3]

Textbook Question

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→∞ (log₂ x - log₃ x)

Textbook Question

{Use of Tech} Write the formula for Newton’s method and use the given initial approximation to compute the approximations x₁ and x₂.


f(x) = x² - 6; x₀ = 3

Textbook Question

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.


∫ (5s + 3)² ds

Textbook Question

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.


lim_x→0 csc 6x sin 7x