Problem 4 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Find \(f^{\prime}(x) .\) Strategize to minimize your work. For example, \(\frac{x^{2}+3}{3 x}\) does not require the Quotient Rule. \(\frac{x^{2}+3}{3 x}=\frac{x}{3}+\frac{1}{x}=\frac{1}{3} x+x^{-1} .\) This is simpler to differentiate. $$ f(x)=\frac{1}{x^{2}+4} $$

Step-by-Step Solution

Answer

The derivative of the function \(f(x) = \frac{1}{x^{2}+4}\) is \(f'(x) = \frac{-2x}{(x^{2}+4)^{2}}\).

Step 1: Recall the Quotient Rule

The quotient rule states that if one has a function that can be written as the quotient of two other functions, say \(f(x) = \frac{g(x)}{h(x)}\), then the derivative of \(f(x)\) can be found using the formula: \[f'(x) = \frac{g'(x)h(x) - g(x)h'(x)} {[h(x)]^2}\]

Step 2: Identify g(x) and h(x)

Write the given function as a ratio of two simpler functions, i.e., \(f(x) = \frac{g(x)}{h(x)}\), where \(g(x)=1\) and \(h(x)={x^{2}+4}\). Compute the derivative of \(g(x)\) and \(h(x)\), denoted as \(g'(x)\), and \(h'(x)\), respectively. In this case \(g'(x) =0\) and \(h'(x) = 2x\).

Step 3: Apply the Quotient Rule

Substitute the values of \(g(x), g'(x), h(x),\text{ and } h'(x)\) into the quotient rule. This results in: \[f'(x) = \frac{(0)\cdot(x^{2}+4) - 1\cdot(2x)} {(x^{2}+4)^{2}} \] Simplify the numerator and denominator separately: \[f'(x) = \frac{-2x}{(x^{2}+4)^{2}}\]

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that involves computing the derivative of a function. It tells us how a function's value changes in response to changes in its input. Essentially, differentiation provides the rate at which function values are changing at any given point. This process is crucial in various fields such as physics, engineering, and economics because it helps identify the speed or rate of change in different contexts.

To differentiate a simple function, we often apply basic rules of differentiation such as the power rule, product rule, and quotient rule. The power rule states that the derivative of a function like \(x^n\) is \(nx^{n-1}\). Nevertheless, when we encounter more complex functions, breaking them down into simpler components before applying differentiation rules can make the task easier and reduce errors.

Derivative of a Function

The derivative of a function at a given point is the slope of the tangent line to the graph of the function at that point. Graphically, this slope represents how steep the graph is rising or falling at that point. In mathematical terms, the derivative is a way of denoting the instantaneous rate of change of the function. For example, in a physical context, if our function represents position with respect to time, the derivative tells us the object's velocity.

Formally, the derivative of a function \( f \) with respect to its variable \( x \) is expressed as \( f'(x) \) or \( \frac{df}{dx} \). There are several notations for derivatives, and the choice of notation often depends on the context of the problem being solved.

Simplifying Expressions Before Differentiation

Simplifying expressions before differentiation is an advantageous strategy that can make finding the derivative more straightforward and help avoid unnecessary complexity. By breaking down a complex function into simpler terms or combining like terms, we can often apply the basic differentiation rules more directly and with less chance of making errors.

For instance, as shown in the exercise, the function \( \frac{x^{2}+3}{3x} \) can be simplified to \( \frac{1}{3}x + x^{-1} \) before differentiating. This means instead of using the more complex quotient rule, one can simply use the power rule, which is easier and faster. The aim of simplifying is to reduce the function to a form where the differentiation becomes almost mechanical, involving fewer steps and simpler calculations.

Finding Derivatives

Finding derivatives is the process of determining the derivative of a function using various rules and techniques of differentiation. For composite functions, the chain rule is applied, whereas for products and quotients, the product rule and quotient rule are used respectively. In practice, being familiar with these rules allows one to tackle a wide range of problems involving rates of change.

In the provided exercise, the quotient rule, which is used for finding the derivative of a function that is the division of two other functions, is applied to the function \( \frac{1}{x^{2}+4} \). The quotient rule formula is \( f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2} \). By identifying \(g(x)\) and \(h(x)\) and their respective derivatives \(g'(x)\) and \(h'(x)\), we use this formula to find the derivative, which, once simplified, gives us \( f'(x) = \frac{-2x}{(x^{2}+4)^{2}} \).