Problem 14 - Calculus and its applications

Question

Differentiate. $$ g(x)=e^{3 x} $$

Step-by-Step Solution

Answer

The derivative of \( g(x) = e^{3x} \) is \( 3e^{3x} \).

Step 1: Understand the Function

The function given is \( g(x) = e^{3x} \). This is an exponential function where the base \( e \) is a constant and the exponent is a linear function of \( x \).

Step 2: Recall the Derivative Rule

The derivative of \( e^{f(x)} \) with respect to \( x \) is given by \( e^{f(x)} \cdot f'(x) \). In our case, \( f(x) = 3x \).

Step 3: Compute the Derivative of the Exponent

Differentiate the exponent \( 3x \) with respect to \( x \). The derivative of \( 3x \) is \( 3 \).

Step 4: Apply the Exponential Derivative Rule

Using the rule from Step 2, the derivative of \( g(x) = e^{3x} \) is \( e^{3x} \cdot 3 \).

Step 5: Simplify the Expression

The expression from Step 4 simplifies to \( 3e^{3x} \). This is the final derivative of the function \( g(x) \).

Key Concepts

Exponential Function

An exponential function is one where the variable appears as the exponent. The base of an exponential function is usually a constant. For example, in the function \( g(x) = e^{3x} \), the base \( e \) is Euler's number, approximately equal to 2.718. This number is special because it naturally arises in many areas of mathematics, often in cases involving continuous growth or decay.
Exponential functions are characterized by their rapid rate of change. This makes them very different from polynomials, which grow at slower rates. In our case, the exponential function involves \( 3x \) as an exponent on \( e \).

• Exponential functions are commonly used to model growth processes, like population growth or compound interest.
• The base \( e \) is used when dealing with natural growth processes, meaning the rates that appear in real-life natural phenomena.

Understanding exponential functions is crucial as they frequently appear in calculus and various real-world applications. They demonstrate how quickly quantities can increase or decrease in different situations.

Derivative Rule

Differentiation is a fundamental concept in calculus that involves finding the rate at which a function is changing. When working with exponential functions, there's a specific rule for finding derivatives. The derivative rule for an exponential function like \( e^{f(x)} \) is given by:
\[ \frac{d}{dx}[e^{f(x)}] = e^{f(x)} \cdot f'(x) \]
This rule combines two important mathematical processes: recognizing the exponential function and applying the chain rule, which allows us to handle compositions of functions.

• The outer function is the exponential function itself, \( e^{f(x)} \).
• The inner function is the exponent, \( f(x) \), which in the given problem is \( 3x \).
• Differentiate the inner function to get \( f'(x) \), in this case, \( 3 \).

The derivative rule tells us to multiply the exponent's derivative, \( 3 \), by the entire exponential expression \( e^{3x} \). Thus, this results in \( 3e^{3x} \). This efficient method allows us to quickly and accurately derive exponential functions without excessive effort.

Linear Function

A linear function is one of the simplest types of functions in mathematics, expressed in the form \( f(x) = ax + b \), where \( a \) and \( b \) are constants. These functions graph as straight lines and have a constant rate of change.
In the context of our problem, the exponent in the function \( g(x) = e^{3x} \) is a linear function \( 3x \). This factor is linear because it fits the general form \( ax + b \) with \( b = 0 \) and \( a = 3 \).

• The simplicity of linear functions makes them easy to differentiate.
• For linear functions, the derivative is simply the slope of the function, which is the coefficient of \( x \) in \( ax + b \).

In our specific example, the derivative of the linear function \( 3x \) is \( 3 \). This is a straightforward computation that plays a critical role when applying the derivative rule to exponential functions with linear exponents. Linear functions serve as building blocks for more complex mathematical models and are essential in understanding more intricate calculus concepts.