Problem 18 - Calculus an Applied Approach

Question

Find the derivative of the function. $$ h(x)=x^{5 / 2} $$

Step-by-Step Solution

Answer

The derivative of the function \(h(x) = x^{\frac{5}{2}}\) is \(h'(x) = \frac{5}{2}x^{\frac{3}{2}}\).

Step 1: Identify the form of the function

The function \(h(x) = x^{\frac{5}{2}}\) is in the form \(f(x) = x^n\), where \(n = \frac{5}{2}\). This means that the power rule of derivatives can be used here.

Step 2: Apply the power rule

The power rule of derivatives tells us that the derivative of \(x^n\) is \(f'(x) = nx^{n-1}\). Applying this rule to our function gives us \(h'(x) = \frac{5}{2}x^{\frac{5}{2} - 1}\).

Step 3: Simplify the expression

Subtracting 1 from the exponent, we get \(h'(x) = \frac{5}{2}x^{\frac{3}{2}}\). So, the derivative of the function \(h(x) = x^{\frac{5}{2}}\) is \(h'(x) = \frac{5}{2}x^{\frac{3}{2}}\). This is the final answer.

Key Concepts

Power Rule in Differentiation

The power rule is one of the most fundamental and widely-used techniques in calculus for finding the derivative of a function. The power rule applies to functions that are in the form of \( f(x) = x^n \), where \( n \) is any real number. This concept simplifies the process of differentiation, making it easier to determine the rate of change of power functions. The formula is straightforward: if \( f(x) = x^n \), then the derivative, noted as \( f'(x) \), is given by:

• \( f'(x) = nx^{n-1} \)

This means you multiply the exponent \( n \) with \( x^{n-1} \), effectively reducing the power of \( x \) by one.For example, if we have a function \( h(x) = x^{\frac{5}{2}} \), using the power rule, we find its derivative as \( h'(x) = \frac{5}{2}x^{\frac{5}{2} - 1} = \frac{5}{2}x^{\frac{3}{2}} \). This simple yet powerful rule helps in efficiently computing derivatives in many calculus problems.

Calculus - The Study of Change

Calculus is a branch of mathematics focused on the study of change and motion. It's divided into two major parts: differentiation and integration. While integration deals with finding the total accumulation of quantities, differentiation is concerned with how quantities change.The origins of calculus lie in the need to solve complex problems in physics and engineering, such as calculating rates of change and areas under curves. Its applications extend beyond mathematics to natural sciences, economics, and even social sciences.In the context of this exercise, calculus helps us understand how the function \( h(x) = x^{\frac{5}{2}} \) behaves as \( x \) changes. By deriving \( h'(x) = \frac{5}{2}x^{\frac{3}{2}} \), we can better grasp the instantaneous rate of change, which is crucial for analyzing dynamic systems.

Differentiation - Understanding Function Behavior

Differentiation is the mathematical process of finding the derivative of a function, which tells us how the function's output changes with respect to a change in input. It's like finding the slope of the tangent line at any point on the function's curve.In simpler terms, differentiation reveals the rate at which something is happening. When you differentiate a function, you obtain another function that predicts changes. This can include predicting speed from a position function or even understanding growth trends in populations.For the function \( h(x) = x^{\frac{5}{2}} \), differentiation using the power rule shows us that \( h'(x) = \frac{5}{2}x^{\frac{3}{2}} \). This result indicates how rapidly the values of \( h(x) \) are changing at any given point \( x \). Analyzing such derivatives provides valuable insights into the behavior of functions and systems over time.