Problem 11 - Technical Mathematics with Calculus

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 Negative Exponent. $$y=x^{-5}$$

Step-by-Step Solution

Answer

\(y' = -5x^{-6} = -\frac{5}{x^6}\)

Step 1: Apply Power Rule for Differentiation

The power rule for differentiation states that the derivative of a function of the form \(f(x) = x^n\) is \(f'(x) = nx^{n-1}\). Here, \(n = -5\), so applying the power rule will give us the derivative.

Step 2: Calculate the Derivative

Following the power rule, the derivative \(y'\) is given by \(y' = (-5)x^{-5-1} = -5x^{-6}\).

Step 3: Simplify the Derivative Expression

We can express the derivative with a positive exponent in the denominator to get \(y' = -5x^{-6} = -\frac{5}{x^6}\).

Step 4: Verify the Result Using a Calculator

Enter the function \(x^{-5}\) into a calculator that can compute derivatives, and check that the output is \(-\frac{5}{x^6}\).

Key Concepts

Derivative Calculation

Understanding derivative calculation is crucial in calculus. A derivative represents how a function changes as its input changes. It's essentially the rate at which the function's output is changing at any given point. The process of finding this rate of change is called differentiation.

In the given exercise, we apply the power rule, which is one of the simplest and most useful rules for differentiation. It directly gives us the derivative of power functions, where the function is of the form y = x^n. The rule is expressed in mathematical notation as f'(x) = nx^{n-1}, where f'(x) denotes the derivative of f(x), and n is a real number. Once the power rule is applied, if the function's power is negative, as in the exercise x^{-5}, we follow the same rule. The negative exponent doesn’t change the power rule’s application, but it does influence the simplification step that usually follows the differentiation process.

Negative Exponents

A negative exponent indicates that the base is on the wrong side of a fraction and needs to be flipped to the other side to evaluate it. For example, x^{-1} is the same as 1/x.

When differentiating a function with a negative exponent, such as y = x^{-5} in our exercise, after applying the power rule, you'll end up with another negative exponent. It is often desirable to express the result using a positive exponent, though this is mainly for aesthetic reasons and to avoid confusion when further manipulating the expression. This involves transforming x^{-6} into 1/x^6, which is its equivalent positive exponent form. Simplifying the derivative this way makes it clearer that as x grows larger, the output of the derivative gets smaller, since it is dividing by a larger and larger number.

Mathematical Notation

Mathematical notation is a language that allows mathematicians to communicate complex ideas efficiently. It's essential for students to learn this language to read and write mathematics effectively.

The notation used in calculus for derivatives can vary. The 'prime' notation f'(x) is one of the most common and is used to denote the derivative of a function f with respect to x. However, when dealing with more complex differentiation, the Leibniz notation dy/dx is often preferred as it clearly indicates which variable is changing.

In our textbook solution, the prime notation is used primarily: the function y = x^{-5} has its derivative notated as y' or f'(x). This language, this notation, is crucial for communicating the concept of derivatives and for performing calculations that involve them. It may seem daunting at first, but with practice, you'll learn to read and write it as fluently as your native language.