Problem 331 - Calculus Volume 1

Question

For the following exercises, find \(f^{\prime}(x)\) for each function. $$f(x)=x^{2} e^{x}$$

Step-by-Step Solution

Answer

\( f'(x) = e^x (x^2 + 2x) \)

Step 1: Identify the Functions

We need to find the derivative of the function \( f(x) = x^2 e^x \). Here, the function is a product of two parts: \( u(x) = x^2 \) and \( v(x) = e^x \).

Step 2: Apply the Product Rule

The product rule states that the derivative of a product \( u(x)v(x) \) is \( u'(x)v(x) + u(x)v'(x) \). We will apply this rule to find \( f'(x) \).

Step 3: Differentiate \( u(x) = x^2 \)

Find the derivative of \( u(x) = x^2 \) with respect to \( x \). Derivative is \( u'(x) = 2x \) because the power rule \( \frac{d}{dx} x^n = nx^{n-1} \) applies here.

Step 4: Differentiate \( v(x) = e^x \)

Find the derivative of \( v(x) = e^x \) with respect to \( x \). The derivative is \( v'(x) = e^x \) since the derivative of \( e^x \) is \( e^x \).

Step 5: Substitute into the Product Rule Formula

Using the derivatives from Steps 3 and 4, substitute into the formula: \( f'(x) = u'(x)v(x) + u(x)v'(x) \). This gives us: \( f'(x) = 2x \cdot e^x + x^2 \cdot e^x \).

Step 6: Simplify the Expression

The expression can be simplified by factoring out \( e^x \): \( f'(x) = e^x (2x + x^2) \). Thus, \( f'(x) = e^x (x^2 + 2x) \).

Key Concepts

Product Rule in Calculus

The product rule is a guideline in calculus used when differentiating products of two functions. Imagine you have a function, like our example, which is composed of two separate parts multiplied together: \( u(x) \) and \( v(x) \). To find the derivative, or the rate at which the function changes, we cannot simply take the derivative of each part and multiply them. Instead, the product rule shows us how to combine the derivatives of these functions.

• The formula is: \( (uv)' = u'v + uv' \).
• This means we differentiate the first function, multiply by the second, add the original first function multiplied by the derivative of the second function.

The key idea is this mixing of original functions with their derivatives, ensuring all paths of change between functions are considered. This is why the product rule is indispensable when the function is a product, like \( x^2 \cdot e^x \). Each part of the product rule formula has a physical interpretation of how changes in one function impact the overall product, combining their contributions effectively.

Understanding the Power Rule

The power rule is one of the simplest rules for differentiation, yet incredibly powerful. It allows us to quickly find the derivative of functions that have the form \( x^n \), where \( n \) is any real number. The power rule states:

• If \( f(x) = x^n \), then the derivative, \( f'(x) = nx^{n-1} \).

For example, in the function \( u(x) = x^2 \), applying the power rule gives us the derivative \( u'(x) = 2x^1 \), which simplifies to \( 2x \). This transformation is simply a result of bringing down the power as a coefficient and reducing the original exponent by one.
The power rule works seamlessly and makes differentiation straightforward because of its consistency. Whenever you see a simple power of \( x \), the power rule should immediately come to mind as a quick and reliable way to find derivatives.

Exponential Function Differentiation

Exponential functions are a unique category in mathematics characterized by their constant growth rates. A common exponential function is \( e^x \), where \( e \) is the base of natural logarithms, approximately equal to 2.718. When it comes to differentiation, exponential functions have a special property:

• The derivative of \( e^x \) is simply \( e^x \) itself.

This means that exponential functions maintain their form even after differentiation, which is why they are so important in calculus and applications involving growth and decay. For \( v(x) = e^x \), the derivative \( v'(x) = e^x \) reflects this property.
This self-similarity in derivatives makes them predictable and easy to work with, especially in equations and models demonstrating natural growth processes, such as populations, radioactive decay, or economic growth models.
Exponential functions exemplify how some natural laws remain unchanged, showing both elegance in mathematics and nature.