Problem 42 - Applied Calculus

Question

Decide which function is an antiderivative of the other. $$f(x)=\frac{2}{3} e^{3 x} ; g(x)=2 e^{3 x}$$

Step-by-Step Solution

Answer

\( f(x) \) is an antiderivative of \( g(x) \).

Step 1: Identify the Derivative of g(x)

We start by taking the derivative of \( g(x) = 2e^{3x} \). Using the chain rule, we get:\[g'(x) = 2 \cdot 3e^{3x} = 6e^{3x}\]

Step 2: Relate f(x) to the Derivative

Now, let's see if \( f(x) = \frac{2}{3} e^{3x} \) could serve as a derivative to relate to 6e^{3x}. Taking the derivative of \( f(x) \):\[f'(x) = \frac{2}{3} \cdot 3e^{3x} = 2e^{3x}\]

Step 3: Determine If One is an Antiderivative of the Other

We've calculated that \( f'(x) = 2e^{3x} \), which matches \( g(x) \). Therefore, \( f(x) \) is an antiderivative of \( g(x) \).

Step 4: Conclusion on Antiderivative Relationship

The function \( g(x) = 2e^{3x} \) is the derivative of \( f(x) = \frac{2}{3} e^{3x} \), confirming that \( f(x) \) is indeed the antiderivative of \( g(x) \).

Key Concepts

Chain Rule

In calculus, the chain rule is a fundamental tool used for finding the derivative of composite functions. When you have a function stuck inside another function, the chain rule helps in differentiating these inside-out combinations. To visualize it, imagine we have a function such as \(e^{3x}\). Here, \(3x\) is the inside function, often referred to as \(u\), and the exponential part, \(e^{u}\), is the outside function.

The chain rule states that the derivative is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. Mathematically, it is expressed as:
\[\frac{d}{dx}[f(g(x))] = f'(g(x)) \, g'(x)\]

In our example of differentiating \(g(x) = 2e^{3x}\), the outer function is the exponential function, \(e^u\), and the inner function is \(3x\). Applying the chain rule gives us:

• Take the derivative of \(e^{3x}\) with respect to \(3x\) which is still \(e^{3x}\).
• Multiply by the derivative of \(3x\), which is \(3\).

This results in \(g'(x) = 6e^{3x}\). This simple rule is a cornerstone for understanding derivatives of complex functions.

Derivative

A derivative represents the rate at which a function is changing at any given point, essentially providing the slope of the function at that exact position. It's a foundational concept in calculus used to determine instantaneous rates of change.

For example, when calculating the derivative of the function \(f(x)\) with respect to \(x\), symbolically expressed as \(f'(x)\), you are finding how \(f(x)\) changes as \(x\) changes.

In the context of the given functions \(f(x) = \frac{2}{3}e^{3x}\) and \(g(x) = 2e^{3x}\), knowing the derivative helps determine if one function serves as the antiderivative of another. Let's focus on \(f'(x)\):

• The derivative of the exponential component \(3x\) is \(3\).
• Multiplying through by the constant \(\frac{2}{3}\), we calculate \(f'(x) = 2e^{3x}\).

This calculation clearly shows that \(f'(x)\) equals \(g(x)\), securing the connection between these two functions as derivatives and antiderivatives.

Exponential Functions

Exponential functions have the form \(f(x) = a \cdot e^{bx}\), where \(e\) is the mathematical constant approximately equal to 2.718, \(a\) is a constant coefficient, and \(b\) indicates the rate of growth or decay. These functions describe processes like population growth, radioactive decay, and continuously compounded interest.

A distinct property of exponential functions is that their derivatives are proportional to the function itself. For instance, the derivative of \(e^{3x}\) is simply \(3e^{3x}\), maintaining the form of the original function.

Looking at our specific functions, \(f(x) = \frac{2}{3}e^{3x}\) and \(g(x) = 2e^{3x}\), both embody exponential characteristics because of \(e^{3x}\). They show how scaling the function by a constant factor directly affects the function’s rate of change rather than its exponential form.

This distinct characteristic simplifies the computational process when calculating derivatives and antiderivatives, making exponential functions a pivotal part of calculus.