Question

(a) Use the Quotient Rule to differentiate the function. (b) Manipulate the function algebraically and differentiate without the Quotient Rule. (c) Show that the answers from \((\mathrm{a})\) and \((\mathrm{b})\) are equivalent. $$g(x)=\frac{x^{3}-2 x^{2}}{2 x^{2}}$$

Short answer

Both differentiation methods yield the derivative \( \frac{1}{2} \). The solutions are equivalent.

Step-by-step solution

Identify the Function for Differentiation

We are given the function \( g(x) = \frac{x^3 - 2x^2}{2x^2} \). The task is to differentiate this function: first using the Quotient Rule, then by simplifying the expression and differentiating without the Quotient Rule.

Differentiate Using the Quotient Rule

The Quotient Rule states that if you have a function \( y = \frac{u}{v} \), then its derivative is \( \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \). Here, \( u = x^3 - 2x^2 \) and \( v = 2x^2 \). Compute \( \frac{du}{dx} = 3x^2 - 4x \) and \( \frac{dv}{dx} = 4x \). Substitute into the Quotient Rule:\[ \frac{d}{dx}\left( \frac{x^3 - 2x^2}{2x^2} \right) = \frac{2x^2(3x^2 - 4x) - (x^3 - 2x^2)(4x)}{(2x^2)^2} \]Simplify the expression to find the result.

Simplify the Quotient Rule Derivative

After substituting the derivatives in Step 2:\[ \frac{2x^2(3x^2 - 4x) - (x^3 - 2x^2)(4x)}{4x^4} = \frac{6x^4 - 8x^3 - 4x^4 + 8x^3}{4x^4} \]Simplify further:\[ \frac{2x^4}{4x^4} = \frac{1}{2} \]Thus, \( \frac{d}{dx}g(x) = \frac{1}{2} \).

Simplify the Function Algebraically

Divide each term in the numerator of \( \frac{x^3 - 2x^2}{2x^2} \) by the denominator:\[ g(x) = \frac{x^3}{2x^2} - \frac{2x^2}{2x^2} = \frac{x}{2} - 1 \]

Differentiate Simplified Function

Differentiate \( g(x) = \frac{x}{2} - 1 \) directly:\[ \frac{d}{dx}g(x) = \frac{1}{2}(1) - 0 = \frac{1}{2} \]

Verify Equivalence of Derivatives

Both methods, using the Quotient Rule and algebraic simplification, led to the derivative \( \frac{1}{2} \). Thus, they are equivalent.

Key concepts

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying an equation or expression to make it easier to work with. In our original problem, this step involved the function \( g(x) = \frac{x^3 - 2x^2}{2x^2} \). The goal was to simplify this complex fraction before differentiating. This process helps to reveal clearer forms of the function that can often be differentiated more simply and with less chance of error.

To perform algebraic manipulation on the given function, we divide each term of the numerator by the denominator separately. Consider each term:\

• \( \frac{x^3}{2x^2} = \frac{x}{2} \)
• \( \frac{-2x^2}{2x^2} = -1 \)

This transforms the original complicated function into \( g(x) = \frac{x}{2} - 1 \). This result is easier to handle when calculating derivatives, as it converts our function into a simple linear form. Algebraic manipulation is a powerful tool in calculus and algebra as it reduces potential complexity in differentiation or other operations.

Differentiation

Differentiation is the mathematical technique used to find the derivative of a function, which represents the rate of change of the function with respect to one of its variables. In the context of our example, we first use the Quotient Rule to differentiate the function \( g(x) = \frac{x^3 - 2x^2}{2x^2} \).

The Quotient Rule is particularly useful for functions expressed as a quotient \( \frac{u}{v} \). According to the Quotient Rule, the derivative of the function is calculated as \( \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \). For our specific function:

• \( u = x^3 - 2x^2 \) and \( \frac{du}{dx} = 3x^2 - 4x \)
• \( v = 2x^2 \) and \( \frac{dv}{dx} = 4x \)

Substituting these into the Quotient Rule formula yields:\\[ \frac{2x^2(3x^2 - 4x) - (x^3 - 2x^2)(4x)}{(2x^2)^2} \].

This process reveals the derivative using the given rule, identifying \( \frac{d}{dx}g(x) = \frac{1}{2} \), the same result achieved by differentiating the simplified form of the function. Differentiation is a fundamental tool in calculus, used across various fields to analyze the behavior of functions.

Simplifying expressions

Simplifying expressions is a crucial skill in mathematics, allowing complex expressions to be transformed into to more manageable forms. The simplification process often involves factoring, canceling terms, and reducing fractions.

In this exercise, after manipulating the function algebraically, we simplified the resulting derivative expression. Initially, after applying the Quotient Rule, the derivative expression obtained was:
\[ \frac{6x^4 - 8x^3 - 4x^4 + 8x^3}{4x^4} \].
This expression asked for factoring and simplification to arrive at a simpler outcome:

• Combine like terms: \(6x^4 - 4x^4 = 2x^4\) and \(-8x^3 + 8x^3 = 0\).
• Reduce the fraction: \( \frac{2x^4}{4x^4} = \frac{1}{2} \).

By simplifying, we confirmed that both methods of differentiation provided the same derivative, \( \frac{1}{2} \) for the original function.

Simplifying expressions not only facilitates easier differentiation but also aids in clearer communication of mathematical thoughts and solutions. It's an essential technique for ensuring accurate and efficient calculation in algebra and calculus.