Problem 23 - Calculus and its applications

Question

Differentiate. $$ g(x)=\frac{1}{2} e^{-5 x} $$

Step-by-Step Solution

Answer

The derivative is \( g'(x) = -\frac{5}{2} e^{-5x} \).

Step 1: Recognize the Function

The function given is an exponential function multiplied by a constant. It is written as \( g(x) = \frac{1}{2} e^{-5x} \).

Step 2: Apply Constant Multiple Rule

To differentiate, you can apply the constant multiple rule of derivatives. This rule allows you to take the constant \( \frac{1}{2} \) out of the differentiation process.

Step 3: Differentiate the Exponential Function

Differentiate the exponential function \( e^{-5x} \). The derivative of \( e^{ax} \) is \( ae^{ax} \). Therefore, the derivative here is \( -5e^{-5x} \).

Step 4: Combine the Results

Multiply the derivative from Step 3 by the constant from Step 2: \( \frac{1}{2} \times (-5e^{-5x}) = -\frac{5}{2} e^{-5x} \).

Step 5: Write the Final Derivative

Thus, the derivative of \( g(x) \) is \( g'(x) = -\frac{5}{2} e^{-5x} \).

Key Concepts

Understanding the Constant Multiple Rule

In calculus, the constant multiple rule is a helpful tool when differentiating functions. This rule simplifies the process when a constant is multiplied by a function. The rule itself states that when you have a constant multiplied by a function, you can take the constant outside of the differentiation process. In mathematical terms, for a constant \( c \) and a function \( f(x) \), the derivative is calculated as:

• \( \frac{d}{dx}[c \cdot f(x)] = c \cdot \frac{d}{dx}[f(x)] \)

This means you first find the derivative of the function \( f(x) \) and then multiply the result by the constant \( c \).
This rule is particularly useful when dealing with exponential functions, like in our exercise where the function is multiplied by \( \frac{1}{2} \). Here, \( \frac{1}{2} \) can be treated separately, making differentiation simpler.

What is an Exponential Function?

Exponential functions are a class of mathematical functions where the variable \( x \) appears in the exponent. These functions have the general form \( e^{ax} \), where \( e \) is the base of the natural logarithm, approximately equal to 2.71828. Exponential functions are important in mathematics due to their unique rate of growth and ability to model real-world phenomena such as population growth and radioactive decay.For instance, in our original exercise, the expression \( e^{-5x} \) is an exponential function. The term \( -5x \) in the exponent indicates a rate of exponential decay. This form is essential in understanding how rapidly the function decreases as \( x \) increases.

Recognizing exponential functions is crucial in solving various calculus problems, including differentiation tasks, because they have unique and specific differentiation rules.

The Derivative of an Exponential Function

The derivative of an exponential function \( e^{ax} \) is distinctive and follows a specific rule. When differentiating exponential functions, the key property is that the base \( e \) remains unchanged, which makes them remarkably easy to differentiate.
The rule for differentiating \( e^{ax} \) is that the derivative is \( ae^{ax} \). This rule relies upon multiplying the function by the coefficient of \( x \) in the exponent \( a \).
Returning to our example of differentiating \( e^{-5x} \), we apply this rule. Here, the exponent has a coefficient of \( -5 \), making the derivative \( -5e^{-5x} \).

This consistency in the derivative makes exponential functions reliable and straightforward to work with during differentiation. By using this rule, we can easily find the rate of change of functions involving exponential terms.