Question

Find the derivative of each function. Verify some of your results by calculator. As usual, the letters \(a, b, c, \ldots\) represent constants. Power Function with Fractional Exponent. $$y=7.5 x^{1 / 3}$$

Short answer

The derivative of the function \(y=7.5x^{1/3}\) is \(y'=2.5x^{-2/3}\).

Step-by-step solution

Identify the Function

First, identify the function you are working with. We have the power function with a fractional exponent where the base is the variable x and the exponent is a fraction, which is given by the expression \(y = 7.5x^{1/3}\).

Use the Power Rule

To find the derivative of \(y = 7.5x^{1/3}\), use the power rule for derivatives. The power rule states that if \(y = ax^n\), then \(y' = nax^{n-1}\). Apply this rule to the function.

Compute the Derivative

Following the power rule, multiply the exponent by the coefficient and then subtract 1 from the exponent. So the derivative of \(y = 7.5x^{1/3}\) will be \(y' = \frac{1}{3} \cdot 7.5x^{\frac{1}{3}-1}\).

Simplify the Exponent

Simplify the expression \(\frac{1}{3}-1\) to get \(-\frac{2}{3}\) and then rewrite the derivative as \(y' = \frac{1}{3} \cdot 7.5 \cdot x^{-2/3}\).

Finalize the Derivative

Simplify the constant by multiplying \(\frac{1}{3} \cdot 7.5\) to get \(2.5\), so the derivative is \(y' = 2.5x^{-2/3}\).

Key concepts

Power Rule for Derivatives

Understanding the power rule for derivatives is crucial in calculus, especially when dealing with power functions like in our exercise. This rule simplifies the process of differentiation by providing an easy formula for finding the derivative of a function of the form y = ax^n, where a is a constant and n is a real number.

According to the power rule, the derivative of such a function is obtained by bringing down the exponent as a multiplier in front of the constant and decreasing the original exponent by one. That is, y' = nax^{n-1}. Applying this rule is straightforward and doesn't require the limits definition of derivatives, making it a powerful tool for students to quickly differentiate functions with exponential terms. For the given function y = 7.5x^{1/3}, using the power rule makes finding the derivative an efficient process.

Fractional Exponents

Fractional exponents in calculus often lead to confusion, but they follow the same rules as integer exponents. A fractional exponent denotes a root, so x^{1/n} equates to the nth root of x. Moreover, the power rule for derivatives applies seamlessly to fractional exponents.

To perform differentiation on a function like y = 7.5x^{1/3}, treat 1/3 as a regular exponent. When the derivative is calculated, and you encounter an expression like x^{1/3-1}, convert this to x^{-2/3}, which is equivalent to dividing by x^{2/3}, or in other terms, dividing by the cube root of x squared. Therefore, a clear understanding of fractional exponents will ensure correct application of the power rule in calculus.

Calculus

Calculus is a branch of mathematics that studies how things change. It provides a framework for modeling systems in which there is change, and a way to predict and understand the behavior of those systems. It is divided mainly into two areas: differential calculus and integral calculus.

In the realm of differential calculus, we are concerned with the concept of a derivative, which represents the rate at which a function is changing at any given point. It is a fundamental tool for finding not just the rate of change, but also for optimizing problems and understanding motion. The power rule for derivatives is a testament to the elegance of calculus, offering a shortcut through the more arduous principles of limits and allowing us to find derivatives quickly and accurately.

Differentiation

Differentiation is the process of finding a derivative of a function. This action is foundational in calculus, allowing us to determine the slope of the tangent line to the function's graph at any given point.

Using differentiation techniques, such as the power rule, we can discover how a system described by a function evolves over time. Whether it's to compute the velocity of an accelerating object from its position function or to find the rate at which a variable cost is increasing relative to production, differentiation provides the necessary analytical tools. The clear step-by-step solution to our initial problem, y = 7.5x^{1/3}, is a practical example of differentiation in action, revealing how straightforward the process can be with a solid grasp of the underlying principles.