Question

Why do two different antiderivatives of a function differ by a constant?

Short answer

Question: Explain why two different antiderivatives of a function differ only by a constant. Answer: Two different antiderivatives of a function differ only by a constant because their difference has a derivative equal to zero, which implies that their difference is a constant function.

Step-by-step solution

Definition of Antiderivative

An antiderivative of a function f(x) is a function F(x), such that the derivative of F(x) is equal to f(x). Mathematically, it can be represented as: F'(x) = f(x)

Property of Functions Differing by a Constant

Suppose we have two functions F(x) and G(x) with the same derivative g(x). Then the derivative of the difference between these functions, i.e., F(x) - G(x), would also be g(x). Mathematically, this can be written as: (F(x) - G(x))' = g(x)

Derivative of Constant

Now consider a constant function C. The derivative of a constant function is always 0, regardless of the value of the constant. Mathematically, this can be represented as: C' = 0

Comparing Derivatives and Constants

Since (F(x) - G(x))' = g(x) and C' = 0, if g(x) is equal to zero, it means that the difference between F(x) and G(x) is a constant function C. Thus, we can write: (F(x) - G(x))' = C' From the above equation, since both sides equal 0, we can say that F(x) and G(x) differ by a constant function, or: F(x) - G(x) = C This means that if we consider two different antiderivatives F(x) and G(x) of a function f(x), they will differ only by a constant as shown by the equation above. In conclusion, any two different antiderivatives of a function will differ by a constant because their difference has a derivative equal to zero, which implies that their difference is a constant function.

Key concepts

Function Differentiation

Differentiation is the process of finding the derivative of a function. A derivative represents how a function changes at any given point and is the opposite of taking an antiderivative. When you have a function, say \( f(x) \), its derivative \( f'(x) \) provides the rate at which \( f \) changes with respect to \( x \).

The reason we differentiate a function is mainly to find this rate of change. When you have a position-time graph, for example, the derivative gives you the speed, or how the position changes over time.

In the context of antiderivatives, if we know \( F'(x) = f(x) \), then \( F(x) \) is an antiderivative of \( f(x) \). The relationship here shows that differentiation and integration (finding the antiderivative) are inverse operations. Given \( f(x) \), you can find its antiderivative by reversing the differentiation process, keeping in mind the role of constants that we add to account for any constant shift in the function's value.

Constant of Integration

When dealing with antiderivatives, the constant of integration plays a crucial role. After finding an antiderivative \( F(x) \), it is customary to include a constant, usually denoted as \( C \). This constant represents any constant number that was lost in the process of differentiation because the derivative of any constant is zero. Thus, whether or not the function had a constant to start, differentiation would eliminate it.

• The constant of integration ensures that every possible antiderivative of a function is described.
• It accounts for all vertical shifts of the function graphically, which do not change the rate of change i.e., the derivative.

This means, if you find one antiderivative \( F(x) \) of a function \( f(x) \), any other antiderivative \( G(x) \) of the same function can be written as \( G(x) = F(x) + C \). This shows how they differ only by that constant.

Properties of Derivatives

Understanding the properties of derivatives is essential to comprehend why two antiderivatives differ by a constant. One fundamental property of derivatives is that the derivative of a constant is zero. This means that any constant added or subtracted to a function does not impact its derivative.

Consider two functions, \( F(x) \) and \( G(x) \), such that \( F'(x) = G'(x) \). This implies that their rate of change is identical at every point. Thus, \( F(x) \) and \( G(x) \) can only be different by a constant, which does not affect the derivative. This is because:

• The derivative of \( (F(x) - G(x)) = 0 \), indicating a zero rate of change, which means the difference itself is a constant.
• The constant added to make the difference zero highlights the property of constant derivatives.

In conclusion, the derivative properties explain why, for any two antiderivatives of the same function, the difference is simply a constant, aligning perfectly with the definition and properties of constant derivatives.