Problem 93 - Calculus and its applications

Question

Differentiate. $$ y=\left(e^{3 x}+1\right)^{5} $$

Step-by-Step Solution

Answer

The derivative is \( \frac{dy}{dx} = 15e^{3x}(e^{3x} + 1)^4 \).

Step 1: Identify the Rule to Use

Since the function is a composition of functions, we will use the chain rule for differentiation, which is applied when differentiating a function of another function. The chain rule states that the derivative of a composite function \( f(g(x)) \) is \( f'(g(x)) \cdot g'(x) \).

Step 2: Differentiate the Outer Function

The outer function in \( y = (e^{3x} + 1)^5 \) is \( u^5 \) where \( u = e^{3x} + 1 \). The derivative of \( u^5 \) with respect to \( u \) is \( 5u^4 \).

Step 3: Differentiate the Inner Function

The inner function is \( u = e^{3x} + 1 \). The derivative of \( u \) with respect to \( x \) is \( 3e^{3x} \) because the derivative of \( e^{3x} \) is \( 3e^{3x} \) and the derivative of 1 is 0.

Step 4: Apply the Chain Rule

With the derivatives from Steps 2 and 3, apply the chain rule. According to the chain rule, multiply the derivative of the outer function by the derivative of the inner function: \[\frac{dy}{dx} = 5(e^{3x} + 1)^4 \cdot 3e^{3x} = 15e^{3x}(e^{3x} + 1)^4\].

Step 5: Simplify the Expression

Combine the constants and simplify the expression as needed. The derivative \( \frac{dy}{dx} = 15e^{3x}(e^{3x} + 1)^4 \) is already in its simplest form for this problem.

Key Concepts

Chain Rule

The chain rule is a fundamental concept in calculus that helps us differentiate composite functions. Essentially, it allows us to find the derivative of a function that is composed of two or more functions, like peeling an onion layer by layer. For instance, if we have a function of the form \( f(g(x)) \), the chain rule gives us a way to differentiate it. The rule states:

• Differential of the outer function multiplied by the differential of the inner function.
• That translates to: \( f'(g(x)) \cdot g'(x) \).

Using the chain rule is like "chaining" the derivatives together, which is why it's called so. This rule is particularly useful when functions are nested inside one another, making direct differentiation challenging. Always remember to identify the "inner" and "outer" functions, and differentiate each separately before applying the rule.

Composite Function

A composite function is created when one function is applied to the result of another. In other words, the output of one function becomes the input of another. You can think of these functions as layers of a beautiful cake, where each layer affects the overall taste and structure.

• The function \( y = (e^{3x} + 1)^5 \) is an example of a composite function.
• The "outer" function here is \( (u)^5 \) and the "inner" function is \( u = e^{3x} + 1 \).

When dealing with composite functions, it's important to separate them into their component parts. Like solving a mystery, you need to identify each part before you can solve the whole. Once identified, you can then apply methods such as the chain rule to find the derivative.

Derivative Rules

Understanding derivative rules is like having a toolbox for solving calculus problems efficiently. Each rule caters to certain types of functions, helping simplify the differentiation process.

• Power Rule: For functions of the form \( u^n \), the derivative is \( nu^{n-1} \).
• Exponential Rule: For functions involving \( e^x \), the derivative is simply \( e^x \).

These rules, among others, form the basis for calculating derivatives. In the given exercise, both the power and exponential rules were used:

• The outer function, \( (e^{3x} + 1)^5 \), was differentiated using the power rule, giving \( 5(e^{3x} + 1)^4 \).
• The inner function, \( e^{3x} \), was differentiated using the exponential rule, yielding \( 3e^{3x} \).

Learning these rules enhances your ability to tackle a wide variety of differentiation problems with confidence.