Problem 18 - University Calculus: Early Transcendentals

Question

Find the derivatives of the functions. $$z=\frac{4-3 x}{3 x^{2}+x}$$

Step-by-Step Solution

Answer

\( \frac{dz}{dx} = \frac{9x^2 - 27x - 4}{(3x^2 + x)^2} \)

Step 1: Identify the Function Type

The function given is in the form of a quotient \( z = \frac{u}{v} \), where \( u = 4 - 3x \) and \( v = 3x^2 + x \). This indicates we will use the quotient rule for derivatives.

Step 2: Write Down the Quotient Rule

The quotient rule states that if \( z = \frac{u}{v} \), then \( \frac{dz}{dx} = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} \). We will derive each part separately and then apply this formula.

Step 3: Differentiate the Numerator \( u = 4 - 3x \)

The derivative of \( u \) with respect to \( x \) is \( \frac{du}{dx} = -3 \).

Step 4: Differentiate the Denominator \( v = 3x^2 + x \)

The derivative of \( v \) with respect to \( x \) is \( \frac{dv}{dx} = 6x + 1 \).

Step 5: Substitute into the Quotient Rule

Substitute \( \frac{du}{dx} = -3 \), \( \frac{dv}{dx} = 6x + 1 \), \( u = 4 - 3x \), and \( v = 3x^2 + x \) into the quotient rule: \[ \frac{dz}{dx} = \frac{(3x^2 + x)(-3) - (4 - 3x)(6x + 1)}{(3x^2 + x)^2} \].

Step 6: Simplify the Expression

Perform the algebraic operations to simplify: 1. Compute \((3x^2 + x)(-3) = -9x^2 - 3x\).2. Compute \((4 - 3x)(6x + 1) = 24x + 4 - 18x^2 - 3x\).3. Simplify the numerator: \(-9x^2 - 3x - 24x - 4 + 18x^2 + 3x = 9x^2 - 27x - 4\).4. The derivative is \[ \frac{dz}{dx} = \frac{9x^2 - 27x - 4}{(3x^2 + x)^2} \].

Key Concepts

Understanding the Quotient Rule

The quotient rule is a vital technique in differential calculus for finding the derivative of a function that is the division of two differentiable functions.
For our given function, which is in the form of \( z = \frac{u}{v} \), where \( u = 4 - 3x \) and \( v = 3x^2 + x \), the quotient rule is necessary to simplify the process.
The rule states:

• If \( z = \frac{u}{v} \), then \( \frac{dz}{dx} = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} \).

This formula helps derive the derivative \( \frac{dz}{dx} \) by accounting for both the numerator and the denominator's derivatives. It ensures that the changes in both parts of the function are considered.
By implementing the quotient rule, we balance the contributions of \( u \) and \( v \) to the overall rate of change. This becomes particularly useful in ensuring accuracy and simplicity in solving derivative problems involving fractions.
Whenever you encounter a division scenario in calculus, the quotient rule should immediately come to mind as a reliable tool.

Steps in Differentiation

Differentiation is the process of finding a derivative, which represents the rate at which a function is changing at any given point.
In our example, the function \( z = \frac{4 - 3x}{3x^2 + x} \) requires us to find the derivatives of both the numerator and the denominator separately before combining them using the quotient rule.
Let's break this down:

• Differentiate the numerator \( u = 4 - 3x \) to find \( \frac{du}{dx} = -3 \).
• Differentiate the denominator \( v = 3x^2 + x \) to get \( \frac{dv}{dx} = 6x + 1 \).

These derivatives represent the instantaneous rate of change of the numerator and the denominator concerning \( x \).
Differentiation allows us to express the behavior of a function in terms of its dynamics, providing insights that are crucial for optimization and understanding motion or change in various contexts.
For effective application of differentiation in functions, expressing the function in simpler forms, like polynomials, may ease the process and lead to swifter solutions.

Mastering Algebraic Simplification

Once you have established the derivatives using the quotient rule, the next crucial step is algebraic simplification.
This makes the expression easier to interpret and use in further calculations. Algebraic simplification is not only about reducing complexity; it also checks for accuracy in combining similar terms and structuring the expression more logically.
Let's go through the process used in our problem:

• Multiply the components: \((3x^2 + x)(-3) = -9x^2 - 3x\), and \((4 - 3x)(6x + 1) = 24x + 4 - 18x^2 - 3x\).
• Combine like terms: Add and subtract terms as needed to streamline the expression \(-9x^2 - 3x - 24x - 4 + 18x^2 + 3x = 9x^2 - 27x - 4\).

The resulting expression \( \frac{9x^2 - 27x - 4}{(3x^2 + x)^2} \) gives a more concise and manageable format of the derivative.
Simplification of algebraic expressions reduces computational burden and enhances clarity, especially in more complicated scenarios._Adjustments like cancelling and re-ordering terms are powerful ways to simplify derivatives and make them more informative and easier to use in practical applications.