Problem 21 - University Calculus: Early Transcendentals

Question

Find \(d y\). $$y=\frac{2 x}{1+x^{2}}$$

Step-by-Step Solution

Answer

The derivative is \( \frac{d y}{d x} = \frac{2(1 - x^2)}{(1+x^2)^2} \).

Step 1: Identify the Problem Type

The problem asks us to find the derivative of a function, specifically \( y = \frac{2x}{1+x^2} \). This is a classic example of a function that requires the quotient rule for differentiation.

Step 2: Recall the Quotient Rule

The quotient rule is used to differentiate functions of the form \( \frac{u}{v} \), where \( u(x) = 2x \) and \( v(x) = 1+x^2 \). The quotient rule states that the derivative \( \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2} \).

Step 3: Differentiate the Numerator and Denominator

First, differentiate the numerator: \( u(x) = 2x \), so \( u'(x) = 2 \). Next, differentiate the denominator: \( v(x) = 1+x^2 \), so \( v'(x) = 2x \).

Step 4: Apply the Quotient Rule

Substitute \( u'(x), u(x), v'(x), \text{ and } v(x) \) into the quotient rule formula:\[\frac{dy}{dx} = \frac{2(1+x^2) - 2x(2x)}{(1+x^2)^2}\]

Step 5: Simplify the Expression

Expand and simplify the expression: \[= \frac{2 + 2x^2 - 4x^2}{(1+x^2)^2}\]This simplifies further to:\[= \frac{2 - 2x^2}{(1+x^2)^2}\]

Step 6: Factor the Numerator

Factor out a \(2\) from the numerator:\[\frac{2(1 - x^2)}{(1+x^2)^2}\]

Key Concepts

Differentiation

Differentiation is a key concept in calculus focusing on how a function changes as its input changes. It is the process of finding the derivative of a function, which provides us with the rate at which the function’s value changes with respect to the change in input. This process is critically important in many fields, such as physics, engineering, and economics, where understanding how variables change concerning one another is valuable.

To differentiate a function, we look at small increments of the variable and observe the resulting changes in the function value.

• The formal definition involves limits, where you consider the behavior of the function as the change in the input approaches zero.
• Practically, rules like the power rule, product rule, and quotient rule help simplify the differentiation process.

In this exercise, the quotient rule is crucial for differentiating the given rational function, illustrating the usefulness of having different strategies in our toolbox for different types of functions.

Function Derivative

A function derivative measures how a function changes when its input changes. This can be thought of as finding the 'slope' of the function at any point, indicating the rate of change.

For example, if you have a function like a curve on a graph, the derivative at a specific point gives you the slope of the tangent to the curve at that point. This slope tells you:

• Whether the function is increasing or decreasing.
• How sharply it is increasing or decreasing.

Using the quotient rule in our example function, where the function is a ratio of two functions, allows us to find the derivative by examining each part separately (numerator and denominator) and how they relate.

The quotient rule helps to methodically break down and calculate the derivative by ensuring the interactions (product and subtraction) between the parts are properly considered.

Calculus

Calculus is a branch of mathematics that deals with continuous change, and it comprises two main branches: differentiation and integration. It was developed to solve complex problems in traditional mathematics that involved motion and change.

Differentiation, as applied in this exercise, is a fundamental part of calculus focused on determining how a function changes at every point, known as the derivative. On the other hand:

• Integration works to find the total accumulation or area under curves.
• Both these concepts enrich our ability to understand and model dynamic systems, whether in physics or economics.

Using calculus and its rules of differentiation allows mathematicians and scientists to describe real-world scenarios precisely, even when they involve constant change or complicated interactions.