Problem 53 - Brief Applied Calculus

Question

Find the derivative of \(\frac{1}{3 x+1}\) in two ways: a. By the Quotient Rule. b. By writing the function as \((3 x+1)^{-1}\) and using the Generalized Power Rule. Your answers should agree. Which way was easier? Remember this for the future.

Step-by-Step Solution

Answer

The derivative using both methods is \( \frac{-3}{(3x+1)^2} \). Both methods yield the same result; the Generalized Power Rule is often easier in this case.

Step 1: Understanding the Function for Derivatives

We are tasked with finding the derivative of the function \( f(x) = \frac{1}{3x+1} \). We will find the derivative using two different methods: the Quotient Rule and the Generalized Power Rule.

Step 2: Using the Quotient Rule

The Quotient Rule states that if we have a function defined by a quotient \( \frac{u}{v} \), its derivative is \( \frac{v \cdot u' - u \cdot v'}{v^2} \). For \( f(x) = \frac{1}{3x+1} \), let \( u = 1 \) and \( v = 3x+1 \). Then, \( u' = 0 \) and \( v' = 3 \). Applying the Quotient Rule, the derivative is:\[ f'(x) = \frac{(3x+1) \cdot 0 - 1 \cdot 3}{(3x+1)^2} = \frac{-3}{(3x+1)^2}. \]

Step 3: Rewriting the Function for the Generalized Power Rule

Rewrite the function \( f(x) = \frac{1}{3x+1} \) as \( f(x) = (3x+1)^{-1} \) to utilize the Generalized Power Rule.

Step 4: Using the Generalized Power Rule

The Generalized Power Rule states that if \( f(x) = (g(x))^n \), then \( f'(x) = n \cdot (g(x))^{n-1} \cdot g'(x) \). Here, \( g(x) = 3x+1 \) and \( n = -1 \). Therefore, \( g'(x) = 3 \). Applying the rule:\[ f'(x) = -1 \cdot (3x+1)^{-2} \cdot 3 = \frac{-3}{(3x+1)^2}. \]

Step 5: Comparing Both Methods

Both methods give the same result: \[ f'(x) = \frac{-3}{(3x+1)^2}. \] This confirms that the derivative is correctly found using either method.

Key Concepts

Quotient Rule

The Quotient Rule is a technique in calculus used to find the derivative of a function that is a quotient of two other functions. In simple terms, think of it as a rule for dividing derivatives. This is particularly useful when dealing with functions where one function is divided by another, like in our exercise where the function is given as \( f(x) = \frac{1}{3x+1} \). The Quotient Rule formula is given by:

• \( \frac{u}{v} \)'s derivative is \( \frac{v \cdot u' - u \cdot v'}{v^2} \)

Here's a friendly reminder:- \( u \) and \( v \) are functions of \( x \)- \( u' \) and \( v' \) are the respective derivatives of \( u \) and \( v \)
For our function, we define:

• \( u = 1 \)
• \( v = 3x+1 \)

By applying the Quotient Rule, the derivative becomes \( f'(x) = \frac{-3}{(3x+1)^2} \), a neat way to handle divisions in calculus.

Power Rule

The Power Rule is one of the simplest and most commonly used rules in differential calculus. It provides a quick way to find the derivative of a function raised to a power. This rule states that if you have a function in the form of \( x^n \), where \( n \) is any real number, then the derivative \( f'(x) \) is given by:

• \( nx^{n-1} \)

In the problem, we converted our original function \( f(x) = \frac{1}{3x+1} \) into \( f(x) = (3x+1)^{-1} \) to employ the Generalized Power Rule. This allows us to treat \( 3x+1 \) like a single variable raised to a power. So:

• \( n = -1 \)
• Apply the Generalized Power Rule: \( n \cdot (g(x))^{n-1} \cdot g'(x) \)

Thus, the derivative becomes: \( f'(x) = \frac{-3}{(3x+1)^2} \). See how the Power Rule makes the process straightforward by tackling powers directly.

Finding Derivatives

Finding derivatives is essential in calculus as it helps understand how a function changes at any given point, often representing the rate of change or slope of the function. Derivatives can be found using different rules and techniques, such as the Quotient Rule and the Power Rule. Whether dealing with simple polynomial functions or more complex fractions and powers:

• Start by identifying the type of function you are working with.
• Choose an appropriate rule: Quotient or Power Rule if applicable.
• Apply the rule methodically, as shown in the exercise.

In our example, both the Quotient Rule and Power Rule yielded the same result for the derivative, illustrating the flexibility and power of these techniques. Understanding when and how to utilize them is important, making derivative finding not just a task but an insightful exploration of rates of change.