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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblema 4.9.19
Capitolo 4, Problema 4.9.19

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


ƒ(x) = eˣ

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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.

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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.
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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.
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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.
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