Problem 30 - Calculus Single Variable

Question

Use the Quotient Rule to compute the derivative of the given expression with respect to \(x .\) $$ \left(x^{2}+1\right) /\left(x^{2}+2\right) $$

Step-by-Step Solution

Answer

The derivative is \( \frac{2x}{(x^2+2)^2} \).

Step 1: Identify the functions

In the expression \( \frac{x^2 + 1}{x^2 + 2} \), identify the numerator and denominator functions. Let \( u(x) = x^2 + 1 \) be the numerator and \( v(x) = x^2 + 2 \) be the denominator.

Step 2: Differentiate the functions

Find the derivatives of the numerator and denominator separately. Compute \( u'(x) = \frac{d}{dx}(x^2 + 1) = 2x \). Similarly, compute \( v'(x) = \frac{d}{dx}(x^2 + 2) = 2x \).

Step 3: Apply the Quotient Rule

The Quotient Rule is given by \( \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2} \). Substitute \( u(x), v(x), u'(x), \) and \( v'(x) \) to get: \[ \frac{d}{dx}\left(\frac{x^2+1}{x^2+2}\right) = \frac{(2x)(x^2+2) - (x^2+1)(2x)}{(x^2+2)^2} \].

Step 4: Simplify the expression in the numerator

Simplify the expression \( (2x)(x^2+2) - (x^2+1)(2x) \): \( 2x^3 + 4x - 2x^3 - 2x \), which simplifies to \( 2x \).

Step 5: Write the final derivative

Substitute the simplified numerator back into the Quotient Rule expression to get the derivative: \[ \frac{2x}{(x^2+2)^2} \].

Key Concepts

Derivatives

Derivatives are a fundamental concept in calculus that allows us to understand how functions change. In simple terms, the derivative of a function at a certain point tells us the rate at which the function's value is changing at that point. This is also known as the function's instantaneous rate of change or the slope of the tangent to the function's graph.
Two main notations are commonly used to express derivatives:

• Lagrange's notation, such as \( f'(x) \), where you see the function's derivative indicated by a prime symbol.
• Leibniz's notation, \( \frac{d}{dx}f(x) \), which highlights the variable with respect to which we are differentiating.

For the function \( y = f(x) \), the derivative \( f'(x) \) gives the change in \( y \) with respect to a tiny change in \( x \). In the context of our exercise, understanding derivatives is key because we are interested in finding the derivative of a rational function using the Quotient Rule.

Differentiation

Differentiation is the process of computing a derivative. It's about applying certain rules to find out how a function behaves as its inputs change. Several rules make differentiation more manageable, one of which is the Quotient Rule.
Differentiation rules simplify the process:

• Power Rule: If \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \).
• Product Rule: For two functions \( u(x) \) and \( v(x) \), the derivative is \( u'v + uv' \).
• Quotient Rule: For \( \frac{u}{v} \), the rule is \( \frac{u'v - uv'}{v^2} \).

In our exercise, differentiation involves the application of the Quotient Rule, which is perfectly suited for derivatives of functions where one function is divided by another, such as the rational function \( \frac{x^2 + 1}{x^2 + 2} \). Mastering differentiation rules allows one to tackle a wide array of calculus problems effectively.

Rational Functions

Rational functions are functions that can be expressed as the quotient of two polynomials. That is, they have the form \( \frac{p(x)}{q(x)} \), where both \( p(x) \) and \( q(x) \) are polynomials, and \( q(x) \) is not zero.
These functions are of great interest because of their complex behavior depending on their numerator and denominator. They can exhibit:

• Vertical asymptotes, where the denominator is zero.
• Horizontal asymptotes, indicating behavior as \( x \) approaches infinity.
• Points of discontinuity, where a possible simplification can lead to undefined behavior.

The rational function \( \frac{x^2 + 1}{x^2 + 2} \) that we look at in the exercise is an example where no simplification occurs to remove the denominator, making it a straightforward candidate for the Quotient Rule. Understanding how these functions work is crucial when differentiating them, as they involve balancing the terms in both the numerator and the denominator when applying differentiation techniques like the Quotient Rule.