Problem 7 - Calculus

Question

Find the derivatives of the functions in exercise. Even if \(n\) is negative or a fraction, the derivative of \(x^{n}\) is \(n x^{n-1}\). $$ x^{n}+x^{-n} $$

Step-by-Step Solution

Answer

The derivative is \(nx^{n-1} - nx^{-n-1}\).

Step 1: Understand the Function

We are given the function \(f(x) = x^{n} + x^{-n}\). Our task is to find the derivative of this function.

Step 2: Differentiate the First Term

For the first term \(x^{n}\), apply the power rule for derivatives. The derivative is \(nx^{n-1}\).

Step 3: Differentiate the Second Term

For the second term \(x^{-n}\), again apply the power rule. The derivative is \(-n x^{-n-1}\).

Step 4: Combine the Derivatives

Combine the derivatives of the two terms to find the derivative of the entire function. Thus, the derivative of \(f(x) = x^{n} + x^{-n}\) is \(nx^{n-1} - nx^{-n-1}\).

Key Concepts

Power Rule

The Power Rule is a fundamental concept in calculus used for finding derivatives of power functions. Whenever you have a function of the form \(x^n\), where \(n\) is any real number, the Power Rule helps you find the derivative efficiently. The rule states that the derivative of \(x^n\) is \(nx^{n-1}\).

This is a super handy rule because it applies to all types of exponents \(n\) - positive, negative, or fractional. It allows us to quickly change the structure of the function to understand how it behaves under small changes. By reducing the power of \(x\) by one and multiplying by the original power \(n\) itself, you get the slope or rate of change of the function.

For example, if \(f(x) = x^3\), then the derivative \(f'(x)\) by the Power Rule is \(3x^{2}\). It's quick and powerful, thus aptly named the Power Rule!

Derivative of Power Functions

Power functions, characterized by a variable base raised to a constant exponent, have derivatives that are found directly using the Power Rule. Generally, for a power function \(f(x) = x^n\), its derivative is given as \(f'(x) = nx^{n-1}\).

This principle holds for any real number \(n\), making it incredibly versatile. Whether you're dealing with a cubic function, a square root (which can be written as a power), or something with a negative exponent, the Power Rule remains consistent.

Here's a quick breakdown:

• Cubic Function (e.g., \(x^3\)): The derivative is \(3x^2\).
• Square Root (e.g., \(\sqrt{x}\) or \(x^{1/2}\)): The derivative is \(\frac{1}{2}x^{-1/2}\).
• Negative Exponent (e.g., \(x^{-4}\)): The derivative is \(-4x^{-5}\).

The derivative represents how the function's value changes as \(x\) changes, which is key to understanding many calculus concepts.

Negative Exponents

Dealing with functions that contain negative exponents can seem tricky at first, but it follows straightforward differentiation rules.

Negative exponents mean the function is essentially a reciprocal power. For instance, \(x^{-n}\) translates to \(1/x^n\) in terms of basic algebra. When applying the Power Rule to function differentiation here, the rule still applies the same way.

Take, for example, the function \(x^{-n}\). Its derivative, using the Power Rule, is \(-nx^{-n-1}\). Notice

• The exponent decreases by 1, consistent with power functions.
• The coefficient becomes negative due to the negative exponent.

Utilizing negative exponents involves a sign change in the derivative, reflecting the inverse nature of the function.

Function Differentiation

Function differentiation is the process of finding the derivative of a function. This measures how a function's output changes as its input changes. In simpler terms, it tells you the "rate of change."

For composite functions or functions consisting of multiple terms, such as \(f(x) = x^n + x^{-n}\), you differentiate each term individually and then sum the derivatives.

Let's break it down:

• First, differentiate \(x^n\) using the Power Rule to get \(nx^{n-1}\).
• Next, differentiate \(x^{-n}\), again with the Power Rule, resulting in \(-nx^{-n-1}\).
• Finally, add these two results: \(nx^{n-1} - nx^{-n-1}\).

This process shows how differentiation gives us a new function representing the slope at any point on the original function, crucial for analyzing function behavior.