Problem 14 - Brief Applied Calculus

Question

Find the derivative of each function by using the Product Rule. Simplify your answers. $$ f(x)=\left(x^{2}+2 x\right)(2 x+1) $$

Step-by-Step Solution

Answer

The derivative using the Product Rule and simplifying is \( f'(x) = 6x^2 + 10x + 2 \).

Step 1: Identify the functions to apply the Product Rule

The Product Rule is defined for functions as \[ \frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x) \]where \( u(x) = x^2 + 2x \) and \( v(x) = 2x + 1 \).

Step 2: Differentiate both functions separately

First, find \( u'(x) \) by differentiating \( u(x) = x^2 + 2x \). The derivative is \[ u'(x) = 2x + 2 \].Next, find \( v'(x) \) by differentiating \( v(x) = 2x + 1 \). The derivative is \[ v'(x) = 2 \].

Step 3: Apply the Product Rule

Now plug the derivatives into the Product Rule:\[ f'(x) = (2x + 2)(2x + 1) + (x^2 + 2x)(2) \].

Step 4: Simplify the expression

First, expand the terms of the expression using distributive property:\[ (2x + 2)(2x + 1) = 4x^2 + 2x + 4x + 2 = 4x^2 + 6x + 2 \].Next, expand the second term: \[ (x^2 + 2x)(2) = 2x^2 + 4x \].

Step 5: Combine like terms

Add the expanded terms together:\[ f'(x) = 4x^2 + 6x + 2 + 2x^2 + 4x \]Combine like terms:\[ f'(x) = 6x^2 + 10x + 2 \].

Key Concepts

Derivatives

Derivatives are a fundamental concept in calculus, often representing the rate at which a function changes at any given point. In simpler terms, the derivative gives the slope of the tangent line to the graph of a function at a specific point. When you hear about finding the derivative, think about how quickly or slowly something is changing.

Some important points to remember about derivatives include:

• The derivative of a constant is zero. This is because constants do not change.
• The derivative of a sum of functions is the sum of their derivatives.
• To find the derivative of a function, you often apply a set of rules, such as the product rule, quotient rule, or chain rule.

It is key to understand which rule to use when calculating derivatives because each rule is suited for different kinds of function combinations. The product rule is one such rule, particularly useful when dealing with the product of two functions.

Calculus

Calculus is a branch of mathematics that deals with continuous change. You've likely heard calculus is divided into two main branches: **differential calculus** and **integral calculus**.

In the case of differential calculus, which deals heavily with derivatives, we focus on examining rates of change and slopes of curves. Calculus allows us to not only find these slopes but also to model and predict related changes.

For example, when we encounter a function that can be expressed as a product of two simpler functions, we use the **product rule** to find its derivative. The product rule is essential because it provides a straightforward method to differentiate such composite functions effectively.

By considering calculus's role in various fields such as physics, economics, and engineering, we recognize how valuable this area of mathematics is in analyzing real-world phenomena.

Mathematical Functions

Mathematical functions are mappings between sets, with each input having a unique output. Functions are written as \( f(x) \), indicating that \( x \) is the variable we input.

Each function can be simple, like \( f(x) = x + 3 \), or more complex, involving multiple operations or compositions of other functions. Understanding the structure and components of functions is vital in calculus, especially when finding derivatives.

In our specific exercise, we deal with two functions that form a product: \( u(x) = x^2 + 2x \) and \( v(x) = 2x + 1 \). The product rule helps us see how each piece contributes to the derivative of the overall function.

To effectively differentiate, we:

• Identify each function involved.
• Calculate the derivative of each function individually.
• Combine using the product rule formula.

By mastering these techniques with mathematical functions, you gain powerful tools for analyzing and interpreting mathematical models.