Problem 19 - Calculus an Applied Approach

Question

Find the derivative of the function. $$ f(x)=4 \sqrt{x} $$

Step-by-Step Solution

Answer

The derivative of the function \(f(x)=4\sqrt{x}\) is \(f'(x) = 2 / \sqrt{x}\)

Step 1: Rewriting the Equation

First, the function \(f(x)=4\sqrt{x}\) can be rewritten in a form more suitable for differentiation using the power rule. The square root of x can be expressed as \(x^{1/2}\), so the function becomes \(f(x) = 4x^{1/2}\).

Step 2: Deriving the equation

Next, the derivative of a function in the form of \(x^n\) is given by \(nx^{n-1}\) (according to the power rule). So, by applying the power rule to the function \(f(x) = 4x^{1/2}\), the derivative \(f'(x)\) will be: \(1/2 * 4x^{1/2-1}\), which simplifies to \(2x^{-1/2}\).

Step 3: Expressing in Standard Form

The result \(f'(x) = 2x^{-1/2}\) can be expressed in a more standard form by observing that \(x^{-1/2}\) is equivalent to \(1/ \sqrt{x}\). So, the function can be expressed as \(f'(x) = 2 / \sqrt{x}\)

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. Simply put, the derivative gives us the rate at which a function is changing at any given point. When we talk about finding a derivative, we're essentially looking at how a small change in the input (usually denoted as \(x\)) affects the output of the function (usually denoted as \(f(x)\)).

To understand differentiation, think of it as finding the "speed" of a function. If you have a function representing distance over time, its derivative tells you the speed or velocity. The process of finding this speed is called "differentiation."

• The derivative is denoted as \(f'(x)\) or \(\frac{df}{dx}\).
• Differentiation involves specific rules and techniques, such as the power rule, product rule, quotient rule, and chain rule, each serving different types of functions.

The primary goal in differentiation is to simplify the function into a form where we can easily identify these rates of change. This often involves rewriting functions in a way that makes them easier to differentiate, which is particularly useful for methods like the power rule.

Power Rule

The power rule is a quick and powerful technique in calculus for finding the derivative of functions of the form \(x^n\), where \(n\) is any real number. Using the power rule makes finding derivatives much simpler, especially when dealing with polynomial functions.

According to the power rule:

• If \(f(x) = x^n\), then the derivative \(f'(x) = nx^{n-1}\).

This means you take the exponent \(n\), bring it down as a multiplier, and then reduce the exponent by one. The power rule can be used for any real number exponent, even fractions or negatives. For example, if you have \(f(x) = x^{1/2}\), applying the power rule gives you \(f'(x) = \frac{1}{2}x^{-1/2}\).

The power rule can be combined with constants effortlessly. If you have a function \(f(x) = 4x^{1/2}\), you would first apply the power rule to \(x^{1/2}\), then multiply the result by 4, as constants in front of functions carry through differentiation.

Square Root Function

A square root function is another common function type you will encounter in calculus, usually written as \(\sqrt{x}\). Square root functions are often transformed into a form suitable for differentiation using the power rule. This is because \(\sqrt{x}\) is equivalent to \(x^{1/2}\).

By rewriting the square root function as \(x^{1/2}\), we can apply the power rule. Differentiating \(f(x) = 4\sqrt{x}\) involves rewriting it as \(f(x) = 4x^{1/2}\).

• Using the power rule on \(x^{1/2}\), we derive \(\frac{1}{2}x^{-1/2}\).
• Multiplying this result by 4 gives us \(2x^{-1/2}\).

Finally, it's helpful to express the derivative in a standard form. Since \(x^{-1/2}\) is the same as \(\frac{1}{\sqrt{x}}\), we rewrite the derivative as \(f'(x) = \frac{2}{\sqrt{x}}\). This transformation is crucial for clarity and ease of interpretation, particularly when evaluating the function's behavior at specific values of \(x\).