Problem 7 - Calculus of a Single Variable

Question

Using the Quotient Rule In Exercises \(7-12,\) use the Quotient Rule to find the derivative of the function. $$ f(x)=\frac{x}{x^{2}+1} $$

Step-by-Step Solution

Answer

The derivative of the function \(f(x)=x/(x^2+1)\) is \(- (x^2 - 1) / (x^2 + 1)^2\)

Step 1: Identify u and v

In this case, \(u(x)\) is the numerator \(x\) and \(v(x)\) is the denominator, which is \(x^2 + 1\).

Step 2: Apply Quotient Rule

Apply the quotient rule which is \((v \cdot u' - u \cdot v') / v^2\). Before applying the rule, calculate the derivative of \(u(x)\) which is \(u'(x) = 1\) and the derivative of \(v(x)\) which is \(v'(x) = 2x\). Now, use these values in the Quotient Rule to calculate the derivative. Substitute \(u\), \(v\), \(u'\), and \(v'\) into the formula, which leads to: \((x^2+1) \cdot 1 - x \cdot 2x\) all divided by \((x^2 + 1)^2 = (x^2 + 1 - 2x^2) / (x^2 + 1)^2\). Simplify to get \(-(x^2 - 1) / (x^2 + 1)^2\)

Step 3: Simplify the Result (if possible)

Simplify the result from Step 2 if possible. In this case, it cannot be simplified any further, so the derivative of the function is \(- (x^2 - 1) / (x^2 + 1)^2\).

Key Concepts

Derivative of a Function

At its core, a derivative represents the rate of change of a function with respect to a variable, typically time or space. When you're figuring out the derivative of a function in calculus, you're hunting for the slope of the tangent line at any point along the function. This concept is a foundational piece of calculus and appears in a variety of scenarios from physics to economics.

The derivative of a simple linear function is a constant, but when functions get more complex, we require techniques like the Quotient Rule to unravel how they change. In the exercise example, we find the derivative of the function by applying this rule, which gives us insight into how the function's rate of change behaves when one function is divided by another.

Calculus

Calculus is the mathematical study of continuous change, and it is divided into two main branches: differential calculus (concerned with the concept of a derivative) and integral calculus (concerned with the concept of an integral).

The use of calculus spans many fields, allowing us to solve problems involving changing quantities. Whether it's calculating the most efficient way to construct a building or determining the speed of a falling object, calculus provides the tools to model and analyze dynamic systems. In our example, calculus helps us find the derivative, or the instantaneous rate of change, of a function with respect to a variable.

Differentiation Techniques

When differentiating functions, we often rely on a number of techniques. The Quotient Rule, as used in the exercise, is just one of them. Others include the Power Rule, the Product Rule, Chain Rule, and rules for differentiating trigonometric, exponential, and logarithmic functions.

Each technique serves a specific type of function or situation. For instance, the Power Rule is for simple polynomials, while the Chain Rule is utilized when you're dealing with composite functions. Effective differentiation often involves combining several of these techniques to tackle more complicated expressions.

Simplifying Expressions

The process of simplifying expressions is a crucial skill in calculus, especially when a raw derivative might not be in its most workable form. Simplifying can reveal properties of the function and make it easier to use in further analysis or graphical interpretation.

Sometimes, simplification may involve factoring, canceling common terms, or expanding expressions. For the function in the exercise, the Quotient Rule leads us to an expression that's already as simple as it can be, so we understand the behavior of the derivative directly from the expression \( -(x^2 - 1) / (x^2 + 1)^2 \).