Problem 116 - Calculus Volume 1

Question

For the following exercises, find \(f^{\prime}(x)\) for each function. $$f(x)=\frac{x^{2}+4}{x^{2}-4}$$

Step-by-Step Solution

Answer

The derivative is \(f'(x) = \frac{-16x}{(x^2 - 4)^2}\).

Step 1: Identify the Function Type

The given function is a rational function, which is the ratio of two polynomials: \(f(x) = \frac{x^2 + 4}{x^2 - 4}\). To find the derivative \(f'(x)\), we must use the Quotient Rule.

Step 2: Write Down the Quotient Rule

The Quotient Rule for differentiation states that if \(f(x) = \frac{u(x)}{v(x)}\), then \(f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}\). Here, \(u(x) = x^2 + 4\) and \(v(x) = x^2 - 4\).

Step 3: Differentiate the Numerator and Denominator

Differentiate \(u(x)\) and \(v(x)\): - \(u'(x) = \frac{d}{dx}(x^2 + 4) = 2x\)- \(v'(x) = \frac{d}{dx}(x^2 - 4) = 2x\).

Step 4: Plug into the Quotient Rule Formula

Substitute \(u(x), v(x), u'(x),\) and \(v'(x)\) into the Quotient Rule: \[ f'(x) = \frac{(2x)(x^2 - 4) - (x^2 + 4)(2x)}{(x^2 - 4)^2} \].

Step 5: Simplify the Expression

Simplify the expression from the previous step:- First, distribute the \(2x\) in each term:\((2x)(x^2 - 4) = 2x^3 - 8x\)\((x^2 + 4)(2x) = 2x^3 + 8x\).- Subtract these: \(2x^3 - 8x - (2x^3 + 8x) = 2x^3 - 8x - 2x^3 - 8x = -16x\).- Therefore, \(f'(x) = \frac{-16x}{(x^2 - 4)^2}\).

Step 6: Finalize the Derivative

The derivative of the function \(f(x) = \frac{x^2 + 4}{x^2 - 4}\) is \(f'(x) = \frac{-16x}{(x^2 - 4)^2}\).

Key Concepts

Quotient Rule

When working with differentiation, the Quotient Rule is a vital tool to master. It helps us find the derivative of a function that is the ratio of two differentiable functions. Suppose you have a function of the form \( f(x) = \frac{u(x)}{v(x)} \).
In such cases, the Quotient Rule tells us that the derivative, \( f'(x) \), is given by:

• Calculate the derivative of the numerator, \( u'(x) \).
• Calculate the derivative of the denominator, \( v'(x) \).
• Next, compute the product \( u'(x)v(x) - u(x)v'(x) \).
• Divide this by the square of the denominator, \( (v(x))^2 \).

These steps might sound tricky at first, but they become manageable with practice. Remember, the numerator of the derivative involves subtracting two products, the first using the derivative of the numerator, and the second using the derivative of the denominator. This approach ensures you calculate the rate of change accurately for complex fractions.

Rational Functions

A rational function is a type of function that represents the ratio of two polynomials. An example is \( f(x) = \frac{x^2 + 4}{x^2 - 4} \). Rational functions are important in calculus and algebra because they often model real-world scenarios where one quantity depends inversely on another.
When differentiating rational functions, you'll frequently use the Quotient Rule, as they involve division of two expressions. Here are some properties and characteristics of these functions:

• They can have vertical asymptotes, which occur when the denominator is zero.
• They may also have horizontal asymptotes or oblique asymptotes, depending on the degrees of the numerator and the denominator.
• Rational functions can be undefined at certain points, specifically where the denominator equals zero.

Understanding and identifying these aspects can help in graphing and mathematically modeling situations using rational functions. Recognizing when to apply the Quotient Rule is crucial for handling these types of functions effectively.

Derivatives

Derivatives form the backbone of calculus, representing the rate at which a quantity changes. They offer deep insights into the behavior of functions. For a function, \( f(x) \), its derivative, denoted as \( f'(x) \), tells us how \( f(x) \) changes as \( x \) varies.
In the context of rational functions, derivatives allow us to understand slopes, rates, and trends, giving a clearer picture of a function's graph:

• They help in finding tangent lines to curves, providing linear approximations.
• The critical points, found where \( f'(x) = 0 \), indicate potential local maxima and minima.
• Derivatives help identify inflection points, places where the curve changes concavity.

Comprehending derivatives and how to compute them for different types of functions like rational functions using guidelines like the Quotient Rule, equips you with powerful tools for analysis and problem-solving in mathematics and beyond. Practice with derivatives brings clarity to predicting and interpreting the behavior of complex functions.