Problem 32 - Brief Applied Calculus

Question

Find the second derivative of each function. $$ \frac{3 x+1}{3 x-1} $$

Step-by-Step Solution

Answer

The second derivative is \( \frac{36}{(3x - 1)^3} \).

Step 1: Identify the function

The function given is \( f(x) = \frac{3x + 1}{3x - 1} \). Our goal is to find the second derivative of this function.

Step 2: First Derivative using the Quotient Rule

To find the first derivative, use the quotient rule which states that if \( f(x) = \frac{u(x)}{v(x)} \), then \( f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} \). Here, \( u(x) = 3x + 1 \) and \( v(x) = 3x - 1 \). Thus, \( u'(x) = 3 \) and \( v'(x) = 3 \). So,\[ f'(x) = \frac{(3)(3x - 1) - (3x + 1)(3)}{(3x - 1)^2} \].

Step 3: Simplify the First Derivative

Expand and simplify the expression from Step 2. \[ f'(x) = \frac{9x - 3 - 9x - 3}{(3x - 1)^2} \]Combine like terms:\[ f'(x) = \frac{-6}{(3x - 1)^2} \].

Step 4: Second Derivative

Differentiate \( f'(x) = \frac{-6}{(3x - 1)^2} \) to find the second derivative. Use the chain rule where if \( g(x) = (3x - 1)^{-2} \), then it is \(-2(3x - 1)^{-3}(3) \). Thus, \[ f''(x) = -6 \times -2 \times (3)(3x - 1)^{-3} \].

Step 5: Final Simplification

Simplify the expression from Step 4:\[ f''(x) = 36(3x - 1)^{-3} \]Convert back to the form of a fraction:\[ f''(x) = \frac{36}{(3x - 1)^3} \].

Key Concepts

Second Derivative

The second derivative of a function is the derivative of the derivative. It's like peeling an onion: you are going deeper into the rate at which things change. If the original function tells you where something is, the first derivative tells you how it is moving—like speed or velocity. The second derivative then tells you how the speed is changing—like acceleration.

In our exercise, we found the first derivative of the function \( f(x) = \frac{3x + 1}{3x - 1} \) by using the quotient rule. After that, the second derivative was calculated to reveal more about the behavior of the original function, particularly how its rate of change itself changes. This is crucial in understanding curvatures and inflection points in graphs. A positive second derivative indicates that the function is concave up, while a negative one indicates it is concave down.

Thus, computing the second derivative helps unravel the deeper intricacies of how a function behaves. It can inform about potential inflection points where the function changes from concave up to concave down, or vice versa.

Quotient Rule

The quotient rule is a handy tool in calculus for finding the derivative of a function that’s the ratio of two differentiable functions. When dealing with fractions, such as in our exercise with \( f(x) = \frac{3x + 1}{3x - 1} \), the quotient rule saves the day.

Mathematically, the quotient rule is expressed as: if you have \( f(x) = \frac{u(x)}{v(x)} \), then the derivative \( f'(x) \) is given by:

• \( f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} \)

Here, both \( u(x) \) and \( v(x) \) are functions of \( x \), and \( u'(x) \) and \( v'(x) \) are their first derivatives.

For our specific function \( f(x) = \frac{3x + 1}{3x - 1} \), we identified \( u(x) = 3x + 1 \) and \( v(x) = 3x - 1 \), both differentiated it, and plugged them into the formula. This method provides a neat and efficient way to find derivatives without resorting to guesswork or estimations.

Chain Rule

The chain rule is another powerful concept in differentiation. It’s used when dealing with composite functions, where one function is nested within another.

The formula for the chain rule is simple yet mighty. If you have a composite function \( h(x) = g(f(x)) \), the derivative \( h'(x) \) is:

• \( h'(x) = g'(f(x)) \cdot f'(x) \)

The essence of the chain rule is taking the derivative of the outer function while keeping the inner function intact, then multiplying by the derivative of the inner function.

In the context of our second derivative calculation for \( f'(x) = \frac{-6}{(3x - 1)^2} \), we treated the inner function \( (3x - 1) \) and applied the chain rule to differentiate \( (3x - 1)^{-2} \). This involved taking the derivative of \( -2(3x - 1)^{-3} \) and then multiplying by the derivative of \( 3x-1 \), which simplifies the process of working through complex fractions and powers more easily. The chain rule makes accessible the intricacies of compound structures in mathematics.