Problem 17 - Thomas Calculus

Question

Find the derivatives of the functions $$ y=\frac{2 x+5}{3 x-2} $$

Step-by-Step Solution

Answer

The derivative is \( y' = \frac{-19}{(3x - 2)^2} \).

Step 1: Identify the Function Type

The function given is a rational function, which can be written as \( y = \frac{u(x)}{v(x)} \), where \( u(x) = 2x + 5 \) and \( v(x) = 3x - 2 \). To find the derivative of the function, we will apply the quotient rule for differentiation.

Step 2: State the Quotient Rule

The quotient rule states that the derivative of a function \( y = \frac{u(x)}{v(x)} \) is given by \( y' = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} \), where \( u'(x) \) is the derivative of \( u(x) \) and \( v'(x) \) is the derivative of \( v(x) \).

Step 3: Find the Derivative of the Numerator

Differentiate \( u(x) = 2x + 5 \). The derivative \( u'(x) \) is simply \( 2 \) since the derivative of \( 2x \) is \( 2 \) and the derivative of the constant \( 5 \) is \( 0 \).

Step 4: Find the Derivative of the Denominator

Differentiate \( v(x) = 3x - 2 \). The derivative \( v'(x) \) is \( 3 \) because the derivative of \( 3x \) is \( 3 \) and the derivative of the constant \(-2 \) is \( 0 \).

Step 5: Apply the Quotient Rule

Use the quotient rule to find the derivative: \[ y' = \frac{(2)(3x - 2) - (2x + 5)(3)}{(3x - 2)^2} \]

Step 6: Simplify the Derivative Expression

Now we simplify the derivative expression:1. Calculate \( (2)(3x - 2) = 6x - 4 \).2. Calculate \( (2x + 5)(3) = 6x + 15 \).3. Subtract to get \( 6x - 4 - (6x + 15) = 6x - 4 - 6x - 15 = -19 \).4. The denominator is \((3x - 2)^2\).Thus, \[ y' = \frac{-19}{(3x - 2)^2} \].

Key Concepts

Quotient Rule

The quotient rule is essential for finding the derivative of a rational function, which is a fraction where both the numerator and the denominator are polynomials. The rule is specifically designed to handle situations when you have to differentiate functions of the form \( y = \frac{u(x)}{v(x)} \).
This is needed because both the numerator and the denominator themselves could change at different rates.
Here’s the formula:

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

• \( u'(x) \) is the derivative of the numerator \( u(x) \).
• \( v'(x) \) is the derivative of the denominator \( v(x) \).

Using the quotient rule ensures that you correctly account for how changes in the numerator and the denominator influence the overall function.

Rational Functions

Rational functions are fractions containing polynomial expressions in both the numerator and the denominator. In simplest terms, they are functions that we can write as \( y = \frac{u(x)}{v(x)} \).
Rational functions are vital in mathematics because they appear frequently in various areas like algebra and calculus.
Some key points about rational functions include:

• They can display different types of behavior, including asymptotes, which are lines that the graph of the function approaches but never actually touches.
• The behavior near the asymptotes is often a beautiful display of mathematical curiosity.
• They can be analyzed by finding their domain, which is all the allowable values of \( x \) such that \( v(x) eq 0 \).

Understanding how to differentiate these functions is crucial for solving many calculus problems.

Differentiation

Differentiation is a fundamental concept in calculus, dealing with how a function changes as its input changes. The process of differentiation allows us to find the derivative of a function, which is a new function showing the rate of change of the original.
In other words, it tells us how fast or slow the original function is increasing or decreasing.
Here are important differentiation points:

• The derivative of a constant is always zero.
• The derivative of \( x^n \) (where \( n \) is a constant) is \( n \times x^{n-1} \).
• To differentiate a sum, simply differentiate each term separately.

In the context of the quotient rule, differentiation is employed to find \( u'(x) \) and \( v'(x) \). Recognizing and applying these basic rules help solve for the derivatives of more complex functions efficiently.

Calculus

Calculus, often thought of as the mathematics of change, embodies two main branches: differentiation and integration. Differentiation, as we discussed, deals with finding the rate of change.
Integration is focused on finding areas under curves.
A few highlights of calculus include:

• Understanding how and why things change within various contexts provides a strong mathematical framework to model natural phenomena.
• It allows the computation of limits, derivatives, and integrals.
• Calculus is the foundation for much of modern science and engineering, which uses its techniques to model and solve real-world problems.

The quotient rule, as applied in this exercise, is part of differential calculus, showing how various parts of the function influence the whole as they individually change.