Problem 37 - Calculus and its applications

Question

Find \(f^{\prime}(x)\). $$f(x)=\frac{4}{x}-x^{3 / 5} $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = -\frac{4}{x^2} - \frac{3}{5}x^{-2/5} \).

Step 1: Differentiate the First Term

The function given is \( f(x) = \frac{4}{x} - x^{3/5} \). We begin by differentiating the first term \( \frac{4}{x} \). This can be rewritten using exponent notation as \( 4x^{-1} \). The derivative of this using the power rule \( \frac{d}{dx}[x^n] = nx^{n-1} \) is \( -4x^{-2} \) or equivalently \( -\frac{4}{x^2} \).

Step 2: Differentiate the Second Term

Differentiating the second term \( x^{3/5} \) using the power rule \( \frac{d}{dx}[x^n] = nx^{n-1} \) yields \( \frac{3}{5}x^{-2/5} \).

Step 3: Combine the Derivatives

Now, we combine the derivatives obtained from both terms. The derivative of the function \( f(x) = \frac{4}{x} - x^{3/5} \) is thus \( f'(x) = -\frac{4}{x^2} - \frac{3}{5}x^{-2/5} \).

Key Concepts

Power Rule

The power rule is a fundamental technique in calculus used for derivative calculation. If you have a function in the form of a power of x, like \( x^n \), the power rule states that the derivative of this function is given by \( nx^{n-1} \). This means you multiply the function by the power and then subtract one from the power.For instance, in the exercise, the term \( \frac{4}{x} \) is rewritten as \( 4x^{-1} \). Using the power rule, its derivative becomes \(-4x^{-2}\), because multiplying by \(-1\) and subtracting one from the exponent of \( x\) yields \(-4x^{-2}\).This powerful rule simplifies the process of differentiation by providing a quick way to handle polynomial expressions and any variable term expressed as a power of \( x \). By applying the power rule, you can easily break down and differentiate complex expressions.

Differentiation Techniques

Differentiation techniques involve several rules and methods to find the derivative of a function efficiently and accurately. The power rule is one of these techniques, but there are others like the product rule, quotient rule, and chain rule, each useful in different scenarios.In the exercise, the power rule was applied. The function was separated into parts, \( \frac{4}{x} \) and \( x^{3/5} \), and each was differentiated separately before combining them. This approach is typical when dealing with functions that are a sum or difference of terms. After differentiation, the derivatives were combined to produce the overall derivative of the function.This methodical separation and differentiation of individual terms illustrate a clear application of differentiation techniques, which are essential for analyzing more complicated functions where components must be handled separately before uniting them together.

Function Analysis

Function analysis refers to examining a function and understanding its behavior by using calculus techniques such as differentiation. By finding the derivative of a function, we gain insight into the rates of change, slopes, or tangents at any point on the curve described by the function.In the context of the exercise, analyzing the function \( f(x) = \frac{4}{x} - x^{3/5} \) meant differentiating it to find \( f'(x) \). The derivative \( f'(x) = -\frac{4}{x^2} - \frac{3}{5}x^{-2/5} \) tells us how the function behaves as \( x \) changes. For instance, it can help determine intervals where the function is increasing or decreasing, assisting in plotting the curve or optimizing solutions in applied problems.Function analysis via derivatives not only helps in understanding graphs but also aids in the theoretical understanding of how variables interact within a given system, making it an invaluable tool in both pure and applied mathematics.