Question

In Exercises \(31-42,\) find \(d y / d x\). $$y=x^{-3 / 5}$$

Short answer

The derivative of the function \(y = x^{-3 / 5}\) with respect to \(x\) is \(\frac{dy}{dx} = -\frac{3}{5}x^{-\frac{8}{5}}\).

Step-by-step solution

Identify the Function

Identify the function and its power. In this case, the function is \(y = x^{-3 / 5}\) and the power \(-3 / 5\).

Apply the Power Rule

Next, apply the Power Rule, which states if \(y = x^n\), then \(\frac{dy}{dx} = n \cdot x^{(n-1)}\). Substitute \(n = -3/5\) into this formula:

Calculate dy/dx

Calculate \(\frac{dy}{dx}\), the derivative of \(y\), with respect to \(x\), by multiplying the power of \(x\) by itself minus one. \(\frac{dy}{dx} = -\frac{3}{5}x^{-\frac{3}{5} - 1} = -\frac{3}{5}x^{-\frac{8}{5}}\)

Key concepts

Derivative Calculation

Derivative calculation is the process of finding the rate at which one variable changes with respect to another. In the context of differential calculus, this usually involves finding the derivative of a function, which provides us with a mathematical snapshot of the function's slope at any given point. Consider the provided exercise where we want to calculate the derivative of the function y = x^{-3/5}.

Following the solution steps, we begin by identifying the function's power, which in this case is -3/5. Understanding that the power signifies the degree to which the variable x is raised is crucial for successfully applying differentiation rules, such as the Power Rule.

Differential Calculus

Differential calculus is one of the two principal branches of calculus, the other being integral calculus. It focuses on the concept of the derivative, which measures how a function changes as its input changes. It's the mathematical foundation for many fields, including physics, economics, and engineering, providing a tool for modeling change and making predictions. In our example, the decrease of the exponent from -3/5 to -8/5 demonstrates how differential calculus allows us to explore the changing behavior of functions.

The Mechanics of Change

In recognizing a function like y = x^n, we understand that the exponent n determines the function's growth rate. As differential calculus teaches us to calculate derivatives, we're equipped to study how even small changes in x can affect the overall value of y, which is especially useful in real-world applications where exact rates of change are necessary.

Applying Power Rule

The Power Rule is a fundamental technique in differential calculus for finding derivatives of functions in the form of x^n, where n is any real number. Applying the Power Rule simplifies the process of differentiation, and it's particularly useful because it applies to any real power of x, not just whole numbers.

To apply the Power Rule, we use the formula \( \frac{dy}{dx} = n \times x^{(n-1)} \) and adjust the exponent by decreasing it by one. In our exercise, the original power -3/5 of x decreases to -8/5 when applying the rule, indicating an increased rate of decrease as x increases. Subtleties like negative exponents mark the importance of understanding how the Power Rule operates under various conditions, including with fractions and negative numbers.

Visualizing the Slope

By calculating -3/5 \times x^{-8/5}, we are effectively determining the slope of the tangent line to the curve of the function y = x^{-3/5} at any point x. This visual representation underscores the essence of the Power Rule in conveying how functions behave at certain points, which is invaluable for analyzing curves in geometry and optimizing functions in calculus.