Problem 32 - Calculus and its applications

Question

Find \(f^{\prime}(x)\). $$ f(x)=0.6 x^{1.5} $$

Step-by-Step Solution

Answer

The derivative \( f'(x) = 0.9x^{0.5} \).

Step 1: Identify the Function and Derivative Rule

The given function is \( f(x) = 0.6x^{1.5} \). To find the derivative, \( f'(x) \), we will use the power rule. The power rule states that if \( f(x) = ax^n \), then \( f'(x) = n imes ax^{n-1} \).

Step 2: Apply the Power Rule

Apply the power rule to \( f(x) = 0.6x^{1.5} \). Here, \( a = 0.6 \) and \( n = 1.5 \). Therefore, the derivative \( f'(x) \) is calculated as follows:\[ f'(x) = 1.5 imes 0.6x^{1.5 - 1} = 1.5 imes 0.6x^{0.5} \].

Step 3: Simplify the Derivative Expression

Calculate \( 1.5 imes 0.6 \) to simplify the expression:\[ 1.5 imes 0.6 = 0.9 \].Therefore, the derivative is:\[ f'(x) = 0.9x^{0.5} \].

Key Concepts

Derivative

In calculus, the derivative represents how a function changes as its input changes. It is the core concept behind understanding motion, rates of change, and many other phenomena in continuous mathematics. The derivative of a function at a particular point gives the slope of the tangent line to the function's graph at that point.
This means how steep the line is at any moment and is a key idea when you're investigating how quantities upscale or downscale.

To find the derivative, there are rules and techniques that simplify the process, one of which is pertinent to our exercise: the Power Rule. By understanding derivatives, you unlock a profound tool to describe and infer the behavior of dynamic systems.

Power Rule

The Power Rule is a straightforward and fundamental method to find the derivatives of polynomials. It specifically applies to functions where the variable is raised to a power, often denoted as \( ax^n \).
This rule makes it incredibly efficient to determine the derivative of such functions without having to use more complex methods or differential quotients.

Here’s how it works:

• If you have a function \( f(x) = ax^n \), the derivative \( f'(x) \) is found by multiplying the exponent \( n \) by the coefficient \( a \), and reducing the exponent by one. This yields \( f'(x) = n \times ax^{n-1} \).

By applying the Power Rule, problems like calculating derivatives of polynomial terms become a matter of quick computation, making it one of the first rules learned in differential calculus.

Function Differentiation

Function differentiation is the process of finding a derivative. Taking the derivative of a function means calculating its rate of change. This could be visualized as finding the slope of a curve at any point.
In our specific exercise, we started with the function \( f(x) = 0.6x^{1.5} \). The goal was to find its derivative \( f'(x) \).

To achieve this, we applied the Power Rule:

• Identify the base form of the function. Here it was \( ax^n \) with \( a = 0.6 \) and \( n = 1.5 \).
• Apply the Power Rule, multiplying \( n \) by \( a \) and subtracting one from \( n \), resulting in \( f'(x) = 1.5 \times 0.6x^{0.5} \).

After simplifying, \( 1.5 \times 0.6 = 0.9 \), we finalized the derivative as \( f'(x) = 0.9x^{0.5} \).
This result gives us the rate of change of the original function and highlights how differentiation is a powerful tool for understanding variable behaviors in mathematical models.