Problem 5 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

For Problems 5 through 9 , differentiate the function given. $$ f(x)=x^{3} e^{x} $$

Step-by-Step Solution

Answer

The derivative of the function \(f(x) = x^{3} e^{x}\) is \(f'(x) = 3x^{2}e^{x} + x^{3}e^{x}\).

Step 1: Identify the two functions

The function given is the product of two functions, namely \(u = x^{3}\) and \(v = e^{x}\).

Step 2: Differentiate each function

Now, differentiate each function separately. The derivative of the function \(u = x^{3}\) is \(u' = \frac{d}{dx}(x^{3}) = 3x^{2}\) and the derivative of the function \(v = e^{x}\) is \(v' = \frac{d}{dx}(e^{x}) = e^{x}\) (since the derivative of \(e^{x}\) with respect to \(x\) is itself).

Step 3: Apply product rule

Applying the product rule for differentiation \(u'v + uv'\), we substitute \(u = x^{3}\), \(v = e^{x}\), \(u' = 3x^{2}\), and \(v' = e^{x}\) into the formula. This gives us \((3x^{2})(e^{x}) + (x^{3})(e^{x})\). Simplifying our derivative, we find \(f'(x) = 3x^{2}e^{x} + x^{3}e^{x}\).

Key Concepts

Differentiation

Differentiation is a key concept in calculus that involves finding the rate at which a function changes. At its core, differentiation lets us determine the slope of the tangent line to the graph of a function at any given point. This is particularly useful when analyzing how functions behave and change over different intervals.

In simple terms, the derivative of a function gives you a mathematical tool to see how a function values' are increasing or decreasing. To differentiate a function, you calculate its derivative, denoted often as \( f'(x) \) if \( f(x) \) is your function. The process involves breaking the function into its basic components and applying rules of differentiation to find the derivative.

• Understand the basic rules: constant rule, sum rule, and power rule.
• Practice by identifying different types of functions for differentiation like polynomials, exponential functions, and products of two functions.

Differentiation becomes much easier with practice and understanding these fundamental rules.

Product Rule

When dealing with the differentiation of a product of two functions, the product rule is your go-to tool. The product rule provides a formula to differentiate expressions where two functions are multiplied together. This is essential because differentiation does not distribute across a product simply.

The product rule states that to differentiate the product \( f(x) = u(x) \, v(x) \), you apply the formula:

• \( (uv)' = u'v + uv' \)

Each term in this formula has a specific meaning:

• \( u' \) is the derivative of the first function, \( u(x) \)
• \( v \) is the original second function
• \( u \) is the original first function
• \( v' \) is the derivative of the second function, \( v(x) \)

Thus, you compute each derivative separately and then multiply according to the rules. This way, you can manage complex expressions efficiently for precise results.

Derivative of Exponential Functions

Exponential functions are unique among other types of functions because their differentiation involves a special, self-similar property. The most common exponential function is \( e^x \), where the base is the mathematical constant \( e \), approximately equal to 2.718.

What's fascinating about the function \( f(x) = e^x \) is that its derivative is itself: \( f'(x) = e^x \). This property simplifies many calculus problems, especially those involving growth models or compound interest calculations.

• Remember that the derivative of \( e^{kx} \) is \( ke^{kx} \).
• Familiarize yourself with the base \( e \) and its natural link to logarithmic differentiation when needed.

This self-similarity and ease of differentiation make exponential functions a cornerstone in calculus, particularly in real-world applications like calculating continuously compounded interest or population growth rates.