Problem 42 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=\frac{5 x^{2}+2 x-6}{3 x-1} $$

Step-by-Step Solution

Answer

\(D_x y = \frac{15x^2 - 10x + 16}{(3x - 1)^2}\).

Step 1: Identify the Function to Differentiate

The given function is a rational function: \[y = \frac{5x^2 + 2x - 6}{3x - 1}\]Our task is to find the derivative \(D_x y\), which is the derivative of \(y\) with respect to \(x\).

Step 2: Apply the Quotient Rule

Recall that the quotient rule is used to differentiate a function of the form \(\frac{u}{v}\):\[D_x \left( \frac{u}{v} \right) = \frac{v \cdot u' - u \cdot v'}{v^2}\]Here, let \(u = 5x^2 + 2x - 6\) and \(v = 3x - 1\).

Step 3: Calculate the Derivatives of the Numerator and Denominator

Now, differentiate \(u\) and \(v\) with respect to \(x\):- For \(u = 5x^2 + 2x - 6\), the derivative \(u' = 10x + 2\).- For \(v = 3x - 1\), the derivative \(v' = 3\).

Step 4: Substitute into the Quotient Rule

Substitute \(u, u', v,\) and \(v'\) into the quotient rule formula:\[D_x y = \frac{(3x - 1)(10x + 2) - (5x^2 + 2x - 6)(3)}{(3x - 1)^2}\]

Step 5: Simplify the Expression

Expand and simplify the expression in the numerator:1. Expand \((3x - 1)(10x + 2) = 30x^2 + 6x - 10x - 2 = 30x^2 - 4x - 2\).2. Expand \( (5x^2 + 2x - 6) \cdot 3 = 15x^2 + 6x - 18 \).3. Subtract the results: \((30x^2 - 4x - 2) - (15x^2 + 6x - 18)\) results in \(15x^2 - 10x + 16\).Finally, the derivative is:\[D_x y = \frac{15x^2 - 10x + 16}{(3x - 1)^2}\]

Step 6: Final Result

Thus, the derivative \(D_x y\) is \[D_x y = \frac{15x^2 - 10x + 16}{(3x - 1)^2}\].

Key Concepts

Quotient Rule

The Quotient Rule is an essential tool in calculus for finding the derivatives of rational functions, which are functions presented as the ratio of two polynomials. This rule is particularly helpful when both the numerator and the denominator are functions of the same variable. The general formula for the quotient rule is expressed as:\[ D_x \left( \frac{u}{v} \right) = \frac{v \cdot u' - u \cdot v'}{v^2} \]
Here, \( u \) and \( v \) are functions of \( x \), while \( u' \) and \( v' \) are their respective derivatives. The Quotient Rule formula shows that the derivative of a quotient is not simply the quotient of the derivatives. Rather, it involves the product of the derivative of the numerator and the original denominator, subtracting the product of the numerator and the derivative of the denominator, all over the square of the original denominator. This method allows us to simplify the process and avoid errors in dealing with more complex rational functions.

Rational Functions

Rational functions are a focal point in calculus differentiation and can be defined as functions that present the ratio of two polynomials. In the exercise at hand, the function given is \( y = \frac{5x^2 + 2x - 6}{3x - 1} \). This function fits the definition of a rational function with a polynomial in both the numerator \( 5x^2 + 2x - 6 \) and the denominator \( 3x - 1 \).
Key characteristics of rational functions include:

• Domain: Rational functions are defined for all real numbers except where the denominator equals zero. Thus, any potential values that make the denominator zero need to be excluded from the domain.
• Asymptotes: There may be vertical asymptotes at the points excluded from the domain (i.e., where the denominator is zero), and horizontal or oblique asymptotes, depending on the degrees of the polynomials in the numerator and denominator.

Understanding these properties helps us analyze the behavior of rational functions and adjust methods in differentiating or solving, especially when applying the quotient rule.

Derivative Calculation

Calculating the derivative of a function, such as a rational function using the quotient rule, is a systematic process. Let's look at the exercise step-by-step to fully understand this process.
First, identify each component of the function to be differentiated. For \( y = \frac{5x^2 + 2x - 6}{3x - 1} \), assign \( u = 5x^2 + 2x - 6 \) and \( v = 3x - 1 \).
Next, compute the derivatives of these components:

• For \( u \), the derivative \( u' = 10x + 2 \) is obtained by differentiating each term separately.
• Similarly, for \( v \), the derivative \( v' = 3 \) is straightforward.

With these derivatives, substitute them back into the quotient rule formula:\[ D_x y = \frac{(3x - 1)(10x + 2) - (5x^2 + 2x - 6)(3)}{(3x - 1)^2} \]
Simplification:
Expand the terms in the numerator:

• \((3x - 1)(10x + 2) = 30x^2 - 4x - 2\).
• \((5x^2 + 2x - 6)(3) = 15x^2 + 6x - 18\).

Subtract these results, leading to \( 15x^2 - 10x + 16 \). Your final differentiated function is then expressed as:
\[ D_x y = \frac{15x^2 - 10x + 16}{(3x - 1)^2} \]
This detailed calculation underscores the benefit of the quotient rule in managing the complexity of rational functions' derivatives.