Problem 10 - Calculus and its applications

Question

Differentiate. $$ y=e^{8 x} $$

Step-by-Step Solution

Answer

The derivative is \( y' = 8e^{8x} \).

Step 1: Identify the Function Type

First, recognize that the function given is an exponential function of the form \(y = e^{u(x)} \), where \( u(x) = 8x \). This will help us apply the differentiation rule for exponential functions.

Step 2: Apply the Exponential Differentiation Rule

Recall that the derivative of \( e^{u(x)} \) is \( e^{u(x)} \cdot u'(x) \). Apply this rule to \( y = e^{8x} \), which requires us to find the derivative of the exponent \( u(x) = 8x \).

Step 3: Differentiate the Exponent

Calculate \( u'(x) \). The derivative of \( 8x \) with respect to \( x \) is simply 8, as it is a linear function.

Step 4: Combine Results

Substitute back into the differentiation rule. The derivative \( y' = e^{8x} \cdot 8 \), which simplifies to \( y' = 8e^{8x} \).

Key Concepts

Exponential Functions

Exponential functions are a unique type of mathematical expression where a constant base is raised to a variable exponent. These functions take the form of \( f(x) = a^{x} \), where \( a \) is a positive constant. The most common base seen in exponential functions is Euler's number, \( e \), which is approximately \( 2.71828 \).
Transforming real-world phenomena like population growth, radioactive decay, and interest accumulation often involves exponential functions.

• For example, the function \( y = e^{8x} \) represents an exponential function where \( e \) is the base and \( 8x \) is the exponent.
• Exponential functions are characterized by their distinctive curve which always grows rapidly, never touching the horizontal axis.

Recognizing an exponential function is the first step in successfully differentiating it. In practical application, if you encounter a problem involving \( e^{u(x)} \), you already know you're dealing with an exponential function.

Derivative Rules

Derivative rules are essential tools in calculus that provide the formulas needed to find the derivative of functions. These rules make differentiating complex functions more straightforward and less time-consuming. Some basic rules include the power rule, product rule, and quotient rule. However, when it comes to exponential functions, there is a specific rule to follow.

For a function of the form \( y = e^{u(x)} \), the derivative can be calculated using:

• Exponential Rule: The derivative of \( e^{u(x)} \) is \( e^{u(x)} \cdot u'(x) \).

This means you need to take the original function \( e^{u(x)} \) and multiply it by the derivative of its exponent \( u(x) \). Using the derivative rules simplifies calculations and ensures accuracy in finding the rate of change of a function.

Chain Rule

The chain rule is a fundamental technique in calculus, particularly useful when differentiating composite functions. A composite function is a function within another function. The chain rule is expressed as:

• \( \text{If } y = f(g(x)), \text{ then } y' = f'(g(x)) \cdot g'(x) \).

Essentially, it's about differentiating from the outer to the inner function. In context, if you have an exponential function like \( y = e^{8x} \), the chain rule is applied as follows:
The outer function is \( e^{u} \) and the inner function is \( u(x) = 8x \). First, differentiate the outer layer as if \( u \) were just a simple variable, which stays \( e^{8x} \). Then, multiply this by the derivative of \( 8x \) (the inner function), which is 8.
Applying the chain rule might initially seem complex but it's just a matter of taking derivatives systematically step-by-step. This method is crucial for managing more complicated calculus problems.