Problem 40 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find \(D_{x} y\) using the rules of this section. $$ y=\frac{5 x-4}{3 x^{2}+1} $$

Step-by-Step Solution

Answer

The derivative is \( D_{x} y = \frac{-15x^2 + 24x + 5}{(3x^2+1)^2} \).

Step 1: Identify the form of the function

The function given is a rational function: \( y = \frac{5x - 4}{3x^2 + 1} \). This is in the form of \( \frac{u(x)}{v(x)} \), where \( u(x) = 5x - 4 \) and \( v(x) = 3x^2 + 1 \). We'll need to apply the quotient rule to differentiate it.

Step 2: Recall the Quotient Rule

The quotient rule is: \( \left( \frac{u}{v} \right)' = \frac{v \cdot u' - u \cdot v'}{v^2} \). We will use this rule for our function where \( u(x) = 5x - 4 \) and \( v(x) = 3x^2 + 1 \).

Step 3: Differentiate the Numerator and Denominator

Differentiate \( u(x) = 5x - 4 \) to get \( u'(x) = 5 \). Differentiate \( v(x) = 3x^2 + 1 \) to get \( v'(x) = 6x \). Now we have: \( u' = 5 \) and \( v' = 6x \).

Step 4: Apply the Quotient Rule

Substitute \( u \), \( v \), \( u' \), and \( v' \) into the quotient rule: \[\left( \frac{5x-4}{3x^2+1} \right)' = \frac{(3x^2+1) \cdot 5 - (5x-4) \cdot (6x)}{(3x^2+1)^2}.\] This will give us the expression for \( D_x y \).

Step 5: Simplify the Expression

Calculate each part: - The first term: \( (3x^2 + 1) \cdot 5 = 15x^2 + 5 \). - The second term: \( (5x - 4) \cdot (6x) = 30x^2 - 24x \). Substitute back to get: \[\frac{15x^2 + 5 - (30x^2 - 24x)}{(3x^2+1)^2}.\] Simplify numerator: \( 15x^2 + 5 - 30x^2 + 24x = -15x^2 + 24x + 5 \).

Step 6: Write the Final Derivative

The final expression of the derivative is: \[D_x y = \frac{-15x^2 + 24x + 5}{(3x^2+1)^2}.\]

Key Concepts

Quotient Rule

When dealing with rational functions in calculus, the Quotient Rule is a handy tool. It helps us find the derivative of functions that are ratios of two other functions. Here’s how it works: if you have a function in the form of \( \frac{u(x)}{v(x)} \), where \( u(x) \) and \( v(x) \) are differentiable functions, then the derivative of this function is given by the formula:\[\left( \frac{u}{v} \right)' = \frac{v \cdot u' - u \cdot v'}{v^2}.\]This formula might look a bit intimidating, but it’s just a series of steps:

• Find the derivative of the numerator, \( u'(x) \).
• Find the derivative of the denominator, \( v'(x) \).
• Substitute these into the quotient rule formula.

This rule helps ensure precision when differentiating complex fractions, making calculus problems involving such functions manageable. Remember, the key is to find the derivatives of both the top and bottom functions, then plug them into the formula.

Derivative

Derivatives are a cornerstone of calculus and provide essential insights into how functions behave. Essentially, the derivative of a function tells you how the function's output changes as its input changes. To find a derivative, you're looking at the rate at which something happens, like how fast a speed increases. When applying the quotient rule, you'll need to differentiate both the numerator and the denominator separately before using those derivatives to find the overall derivative of the rational function.
Differentiating means you compute the derivative, finding \( u'(x) \) and \( v'(x) \) where \( u(x) \) and \( v(x) \) are parts of your rational function. The derivative provides a linear approximation of the function at any given point and helps solve optimization problems or find where the function is increasing or decreasing. It's a fundamental concept, unlocking a deeper understanding of calculus.

Rational Function

A rational function is any function that can be expressed as the ratio of two polynomial functions. In other words, it looks like \( \frac{u(x)}{v(x)} \) where \( u(x) \) and \( v(x) \) are polynomials. Rational functions are common in real-world applications where they model ratios and other division-based calculations.
To differentiate such a function, we use the quotient rule. The numerator and denominator of a rational function are often polynomials, which makes them easier to differentiate using basic rules of derivatives for polynomials. It's crucial to handle both expressions carefully, ensuring every part is accurately differentiated and simplified.

• Numerator: The top part of the fraction function.
• Denominator: The bottom part that influences the behavior of the function.

Rational functions can produce asymptotes in graphs, where the function approaches but never touches certain lines. Understanding and differentiating rational functions is vital for dealing effectively with advanced calculus and real-world mathematical models.