Problem 19 - University Calculus: Early Transcendentals

Question

Find the derivatives of the functions. $$g(x)=\frac{x^{2}-4}{x+0.5}$$

Step-by-Step Solution

Answer

The derivative is \( g'(x) = \frac{x^2 + x + 4}{(x + 0.5)^2} \).

Step 1: Identify Quotient Rule

The function \( g(x) = \frac{x^2 - 4}{x + 0.5} \) is a quotient of two functions. To find the derivative, we'll apply the quotient rule for differentiation, which states that if \( y = \frac{u}{v} \), then \( y' = \frac{u'v - uv'}{v^2} \). Identify \( u = x^2 - 4 \) and \( v = x + 0.5 \).

Step 2: Differentiate the Numerator

Differentiate the numerator \( u = x^2 - 4 \) with respect to \( x \). The derivative \( u' = 2x \) because the derivative of \( x^2 \) is \( 2x \) and the derivative of \(-4\) is \( 0 \).

Step 3: Differentiate the Denominator

Differentiate the denominator \( v = x + 0.5 \) with respect to \( x \). The derivative \( v' = 1 \) because the derivative of \( x \) is \( 1 \) and the derivative of the constant \( 0.5 \) is \( 0 \).

Step 4: Apply Quotient Rule

Using the quotient rule formula \( y' = \frac{u'v - uv'}{v^2} \), substitute \( u' = 2x \), \( v = x + 0.5 \), \( u = x^2 - 4 \), and \( v' = 1 \) into the formula. This gives:\[ g'(x) = \frac{(2x)(x + 0.5) - (x^2 - 4)(1)}{(x + 0.5)^2} \]

Step 5: Simplify the Expression

Simplify the expression obtained in Step 4:1. Calculate \( (2x)(x + 0.5) = 2x^2 + x \).2. Calculate \( (x^2 - 4)(1) = x^2 - 4 \).Substitute these back into the derivative:\[ g'(x) = \frac{2x^2 + x - (x^2 - 4)}{(x + 0.5)^2} \]Simplify the numerator:\[ g'(x) = \frac{2x^2 + x - x^2 + 4}{(x + 0.5)^2} = \frac{x^2 + x + 4}{(x + 0.5)^2} \].

Key Concepts

Quotient Rule

The Quotient Rule is a fundamental rule in calculus used to differentiate functions that are divided by each other, in other words, functions expressed as a quotient. It applies whenever you have a function of the form \( g(x) = \frac{u(x)}{v(x)} \). To find the derivative \( g'(x) \), use the formula:\[ g'(x) = \frac{u'(x) v(x) - u(x) v'(x)}{[v(x)]^2} \]Here, \( u(x) \) is the numerator function, and \( v(x) \) is the denominator function. The formula's premise is to first calculate the derivatives of the numerator \( u'(x) \) and the denominator \( v'(x) \) separately.
- **Step 1:** Differentiate the top function \( u(x) \).- **Step 2:** Differentiate the bottom function \( v(x) \).- **Step 3:** Substitute these derivatives into the quotient rule formula. In this exercise, for the function \( g(x) = \frac{x^2 - 4}{x + 0.5} \), \( u(x) \) is \( x^2 - 4 \) and \( v(x) \) is \( x + 0.5 \). Proceed by differentiating each part before applying the quotient formula.

Differentiation

Differentiation is the process of finding the derivative of a function, which tells us the rate at which the function's value changes as its input changes. It is a fundamental tool in calculus for analyzing and understanding the behavior of functions.
For a simple polynomial function like \( f(x) = x^2 - 4 \), the process of differentiation involves applying basic rules such as the power rule, where the derivative \( d/dx[x^n] \) is \( nx^{n-1} \). Applying this to \( x^2 \), we get \( 2x \), and the derivative of a constant such as \(-4\) is \( 0 \).
- **Step 1:** Apply the power rule to differentiate \( x^2 \), resulting in \( 2x \).- **Step 2:** Recognize that the derivative of a constant is \( 0 \).- **Step 3:** Combine these results for the overall derivative of the numerator and denominator separately.In our exercise, \( u'(x) = 2x \) and \( v'(x) = 1 \) for the denominator \( v(x) = x + 0.5 \), since the derivative of \( x \) is \( 1 \) and the derivative of the constant \( 0.5 \) is \( 0 \).

Simplifying Expressions

Simplifying expressions is crucial in calculus to make derivatives more manageable and interpretable. After finding a derivative using rules like the quotient rule, simplifying involves combining like terms and reducing fractions to their simplest forms. This step is essential for clarity and further analysis of a mathematical problem.
To simplify, follow these steps:- **Step 1:** Perform algebraic operations such as distributive multiplication and combining like terms.- **Step 2:** Wherever possible, cancel common factors in a fraction.In the exercise, after applying the quotient rule, the expression obtained was:\[ g'(x) = \frac{(2x)(x + 0.5) - (x^2 - 4)(1)}{(x + 0.5)^2} \]
- Start by expanding: \( 2x(x + 0.5) = 2x^2 + x \) and keep \( (x^2 - 4)(1) = x^2 - 4 \).- Substitute these into the expression:\[ g'(x) = \frac{2x^2 + x - x^2 + 4}{(x + 0.5)^2} \]
- Simplify the numerator by combining like terms to get:\[ g'(x) = \frac{x^2 + x + 4}{(x + 0.5)^2} \]Simplification ensures that the expression is in its simplest form, which is easier to work with in further calculations or when interpreting the results.