Problem 41 - Calculus and its applications

Question

Find \(f^{\prime}(x)\). $$ f(x)=\frac{x^{3 / 2}}{3} $$

Step-by-Step Solution

Answer

The derivative is \( f^{\prime}(x) = \frac{1}{2}x^{1/2} \).

Step 1: Identify the Function

We begin by identifying the function we need to differentiate: \( f(x) = \frac{x^{3/2}}{3} \). This can also be rewritten for differentiation purposes as \( f(x) = \frac{1}{3}x^{3/2} \).

Step 2: Apply the Power Rule

The power rule of differentiation states that \( \frac{d}{dx} x^n = nx^{n-1} \). We will apply this rule to \( \frac{1}{3}x^{3/2} \). First, differentiate \( x^{3/2} \): \( \frac{3}{2}x^{1/2} \).

Step 3: Differentiate with Constant

Since the function has a constant multiplier \( \frac{1}{3} \), it remains as a coefficient of the derivative from Step 2: \( \frac{1}{3} \times \frac{3}{2}x^{1/2} = \frac{3}{6}x^{1/2} \).

Step 4: Simplify the Expression

Simplify the derivative expression: \( \frac{3}{6}x^{1/2} = \frac{1}{2}x^{1/2} \). Therefore, the derivative of the function is \( f^{\prime}(x) = \frac{1}{2}x^{1/2} \).

Key Concepts

Power Rule

Differentiation is a crucial concept in calculus that allows us to find how a function changes at any given point. The Power Rule is one of the simplest and most commonly used techniques to differentiate functions. This rule applies when we have a power of a variable, such as \( x^n \). The Power Rule states that if \( f(x) = x^n \), then the derivative \( f^{\prime}(x) \) can be found by multiplying the exponent \( n \) by the variable raised to the power of \( n-1 \). Mathematically, it's expressed as: \[\frac{d}{dx} x^n = nx^{n-1}\]This rule is extremely powerful because it simplifies the process of differentiation. Even more complex functions involving various powers of \( x \) can often be simplified to apply the Power Rule directly.
When differentiating a function that includes constants, such as \( \frac{1}{3}x^{3/2} \), the Power Rule still applies. The constant stays the same while you carry out the differentiation on the variable part. This allows for straightforward and quick calculations.

Derivative

The derivative is a foundational concept that measures how a function changes as its input changes. It provides the slope of the tangent line to the function's graph at any particular point. In simple terms, the derivative tells us the rate of change or the velocity of the function.
For example, if you have a function \( f(x) = x^2 \), its derivative \( f^{\prime}(x) = 2x \) shows how quickly \( f(x) \) increases or decreases as \( x \) changes.In the context of the given exercise, the derivative of the function \( f(x) = \frac{1}{3}x^{3/2} \) is found by using the Power Rule. We first focus on the part \( x^{3/2} \), determine its derivative using the Power Rule, and then include any constants as coefficients in the final derivative.
The derivative of \( x^{3/2} \) is \( \frac{3}{2}x^{1/2} \), and when combined with the constant \( \frac{1}{3} \), the resulting derivative is \( f^{\prime}(x) = \frac{1}{2}x^{1/2} \). This result indicates the rate of change of the function at any value of \( x \).

Function Simplification

Simplification is the process of making a mathematical expression as simple as possible. In calculus, we often encounter expressions that can be complex or cumbersome to work with in their original form.Simplifying allows for easier manipulation and calculation.
In the exercise solution, after applying the Power Rule and differentiation, we see that the step involves simplifying the derivative expression.The initial derivative \( \frac{1}{3} \times \frac{3}{2}x^{1/2} \) simplifies first through arithmetic: multiplying the constants \( \frac{1}{3} \times \frac{3}{2} = \frac{3}{6} \), which then simplifies to \( \frac{1}{2} \).
Therefore, the final simplified form of the derivative is \( \frac{1}{2}x^{1/2} \).This simplification not only makes the function easier to interpret but also easier to use if further calculus operations are needed. It breaks down what could appear complex into a more manageable and understandable form, which is essential for problem-solving in calculus.