Problem 8 - Single Variable Calculus

Question

Use the Quotient Rule to find the derivative of each function. \(g(x)=\frac{2 x}{x^{2}+1}\)

Step-by-Step Solution

Answer

The derivative of the given function, using the Quotient Rule, is: \(g'(x) = \frac{-2x^2 + 2}{(x^2 + 1)^2}\).

Step 1: Identify the function components

In our given function \(g(x) = \frac{2x}{x^2 + 1}\), we can identify the top function (numerator) as: \[f(x) = 2x\] and the bottom function (denominator) as: \[g(x) = x^2 + 1\] Now, we need to find the derivatives of these functions, \(f'(x)\) and \(g'(x)\).

Step 2: Find the derivatives of the function components

For function \(f(x) = 2x\), its derivative is: \[f'(x) = 2\] For function \(g(x) = x^2 + 1\), its derivative is: \[g'(x) = 2x\]

Step 3: Apply the Quotient Rule to find the derivative

Now that we have \(f'(x)\), \(g(x)\), and \(g'(x)\), we can plug these into the Quotient Rule formula: \[h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}\] Substitute with the values we found: \[g'(x) = \frac{(2)(x^2 + 1) - (2x)(2x)}{(x^2 + 1)^2}\]

Step 4: Simplify the derivative

Now, we will simplify the expression for \(g'(x)\): \[g'(x) = \frac{2x^2 + 2 - 4x^2}{(x^2 + 1)^2}\] Combine the terms in the numerator: \[g'(x) = \frac{-2x^2 + 2}{(x^2 + 1)^2}\] Finally, the derivative of the given function, using the Quotient Rule, is: \[g'(x) = \frac{-2x^2 + 2}{(x^2 + 1)^2}\]

Key Concepts

Derivatives

Derivatives are a fundamental concept in calculus and are used to determine the rate of change of a function with respect to a variable. For any given function, the derivative provides a way to examine the behavior of the function at any given point. In simpler terms, if you want to know how fast something is changing at a specific moment, you calculate its derivative.

There are several key points to remember about derivatives:

• The notation for derivative can vary; for functions, they can be written as \(f'(x)\) or \(\frac{dy}{dx}\).
• Derivatives of basic functions have specific rules; for example, the derivative of \(x^n\) (where \(n\) is a constant) is \(nx^{n-1}\).
• The derivative tells us about the slope of the tangent to the curve at a particular point.

In the exercise, the derivative of the functions \(f(x) = 2x\) and \(g(x) = x^2 + 1\) were identified as \(f'(x) = 2\) and \(g'(x) = 2x\) respectively. Finding these derivatives is crucial for applying the Quotient Rule.

Calculus

Calculus is the field of mathematics that deals with rates of change and the accumulation of quantities. It has two primary branches: differential calculus, which involves finding derivatives, and integral calculus, which deals with finding integrals and area under curves. The Quotient Rule is a tool within differential calculus and is used to find derivatives of functions that are written as a fraction of two other functions. This is especially useful when dealing with complex rational functions.

The power of calculus lies in its ability to provide precise mathematics to model the real world. Whether we're looking at scientific phenomena, economic trends, or engineering principles, calculus provides the foundation for understanding how things change. In our context, it helps us find the rate of change of a rational function using the Quotient Rule.

Rational Functions

Rational functions are a type of function represented as the ratio of two polynomial functions. They often look like \(\frac{p(x)}{q(x)}\) where both \(p(x)\) and \(q(x)\) are polynomials. Understanding rational functions is essential since many real-world relationships can be modeled through them.

In the exercise, the function \(g(x) = \frac{2x}{x^2 + 1}\) is a rational function. This particular function has a numerator, \(2x\), and a denominator, \(x^2 + 1\). When studying rational functions:

• The denominator must not be zero, as dividing by zero is undefined in mathematics.
• To analyze these functions, it's often vital to simplify them or use techniques like the Quotient Rule to differentiate them.
• They can exhibit complex behaviors such as vertical asymptotes and horizontal or slant asymptotes, influencing their graph shapes significantly.

Rational functions are a key concept within calculus and understanding how to differentiate them using rules like the Quotient Rule is crucial for deeper mathematical analysis.