Problem 49 - Calculus and its applications

Question

Find \(y^{\prime}\) $$ \text { If } f(x)=\sqrt{x}, \text { find } f^{\prime}(4) $$

Step-by-Step Solution

Answer

The value of \( f'(4) \) is \( \frac{1}{4} \).

Step 1: Identify the function and variables

We are given the function \( f(x) = \sqrt{x} \). Our task is to find \( f'(x) \), the derivative of \( f(x) \), and then evaluate it at \( x = 4 \).

Step 2: Differentiate the function

To find the derivative \( f'(x) \), we use the power rule. Recall that \( \sqrt{x} \) can be written as \( x^{1/2} \). The power rule states that if \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \). Applying this rule: \[ \frac{d}{dx}[x^{1/2}] = \frac{1}{2}x^{-1/2} \].

Step 3: Simplify the derivative

The derivative, \( f'(x) = \frac{1}{2}x^{-1/2} \), can be rewritten in a simpler form as \( \frac{1}{2\sqrt{x}} \). This is because \( x^{-1/2} \) is equivalent to \( \frac{1}{\sqrt{x}} \).

Step 4: Evaluate the derivative at x = 4

Substitute \( x = 4 \) into the derivative \( f'(x) = \frac{1}{2\sqrt{x}} \). So \( f'(4) = \frac{1}{2\sqrt{4}} \). Since \( \sqrt{4} = 2 \), it follows that \( f'(4) = \frac{1}{4} \).

Key Concepts

Power Rule

The Power Rule is one of the most basic differentiation rules. It allows us to find the derivative of functions quickly, especially polynomial functions. If you have a variable raised to a power, the Power Rule comes to the rescue. Let's break it down:

• Start with a function of the form \( f(x) = x^n \), where \( n \) is any real number.
• The derivative \( f'(x) \) is found by multiplying \( n \) by \( x^{n-1} \).

For instance, for \( f(x) = x^3 \), the derivative would be \( f'(x) = 3x^{2} \). You bring down the exponent as a coefficient and then subtract one from the original exponent.
In our exercise, we used the Power Rule when finding the derivative of \( \sqrt{x} \), which can be rewritten as \( x^{1/2} \). According to the Power Rule, the derivative becomes \( \frac{1}{2}x^{-1/2} \). This powerful method streamlines the process of differentiation and is widely used in calculus.

Square Root Function

The square root function, denoted as \( \sqrt{x} \), is one of the fundamental functions in mathematics. It is the inverse function of squaring a number and belongs to the family of root functions. Let's look into some key points:

• The expression \( \sqrt{x} \) can be rewritten in exponential form as \( x^{1/2} \). This is particularly useful for differentiation.
• The graph of the square root function is a gentle curve that starts at the origin \((0,0)\) and progresses into the positive quadrant, since every real-valued square root yields a non-negative value.

When differentiating the square root function, converting it to an exponent allows us to apply the differentiation rules with ease. This is why, when determining derivatives, \( \sqrt{x} \) is often seen as \( x^{1/2} \), helping to apply the Power Rule smoothly.

Evaluating Derivatives

Once we've found the derivative of a function, the next step is often evaluating it at a specific point. This step is particularly useful when solving real-world problems, such as finding a slope at a given point or calculating instantaneous rates of change. Here's a simple guide on how to do it:

• First, determine the derivative of the given function. In our example, we found the derivative \( f'(x) = \frac{1}{2\sqrt{x}} \).
• Next, substitute the specific value of \( x \) into the derivative equation. This will give you the slope or rate of change at that particular point. For \( x = 4 \), substitute this into \( f'(x) \) to obtain \( f'(4) = \frac{1}{4} \).

This evaluated derivative provides crucial insight into the behavior of the function at the specified point. It's a way to measure how fast \( f(x) \) is changing precisely at \( x = 4 \), thus providing a snapshot of the function's slope.