Problem 76 - University Calculus: Early Transcendentals

Question

Use the Derivative Quotient Rule to prove the Power Rule for negative integers, that is, $$\frac{d}{d x}\left(x^{-m}\right)=-m x^{m-1}$$ where \(m\) is a positive integer.

Step-by-Step Solution

Answer

The Power Rule for negative integers is proven by applying the Quotient Rule.

Step 1: Recall the Quotient Rule

The derivative quotient rule states: \( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \) where \( u \) and \( v \) are differentiable functions of \( x \).

Step 2: Express \( x^{-m} \) as a Quotient

We write \( x^{-m} = \frac{1}{x^m} \). Here, let \( u = 1 \) and \( v = x^m \).

Step 3: Compute Derivatives of \( u \) and \( v \)

For \( u = 1 \), the derivative \( \frac{du}{dx} = 0 \). For \( v = x^m \), the derivative \( \frac{dv}{dx} = m x^{m-1} \) using the power rule for positive integers.

Step 4: Apply the Quotient Rule

Substitute into the quotient rule: \[\frac{d}{dx} \left( \frac{1}{x^m} \right) = \frac{x^m \cdot 0 - 1 \cdot mx^{m-1}}{(x^m)^2} = \frac{-mx^{m-1}}{x^{2m}}.\]

Step 5: Simplify the Expression

Simplify the expression obtained:\[\frac{-mx^{m-1}}{x^{2m}} = -m \cdot x^{m-1 - 2m} = -m \cdot x^{-m-1}.\]

Step 6: Confirm the Power Rule for Negative Exponents

Thus, we have shown that \( \frac{d}{dx}(x^{-m}) = -mx^{-m-1} \), confirming the Power Rule for negative integers.

Key Concepts

Derivative Quotient Rule

The Derivative Quotient Rule is a fundamental tool in calculus, especially when tackling the differentiation of quotient forms. This rule allows us to find the derivative of a function presented as a ratio of two other functions. Using this rule, if you have a function of the form \( \frac{u}{v} \), where both \( u \) and \( v \) are differentiable, the derivative is expressed as:

• Find the derivative of the numerator \( \frac{du}{dx} \).
• Find the derivative of the denominator \( \frac{dv}{dx} \).
• Apply the formula: \( \frac{d}{dx}\left( \frac{u}{v} \right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \).

Keep in mind, the order of terms matters. If these terms are not properly handled, the calculation will result in errors. Thus, understanding and correctly applying each step is crucial when solving problems utilizing the quotient rule.

Power Rule

The Power Rule is one of the most commonly used differentiation techniques in calculus. It provides a quick method to find the derivative of polynomial functions. Usually, the Power Rule applies to functions of the form \( x^n \), where \( n \) is any real number, and states that:

• To differentiate \( x^n \), multiply by the exponent \( n \).
• Subtract one from the exponent.
• Thus, the derivative is \( nx^{n-1} \).

For example, the derivative of \( x^3 \) is \(3x^2 \). This rule's efficacy shines in its simplicity, allowing students to quickly ascertain derivatives of basic monomials. Understanding how to apply the Power Rule effectively is vital for handling more complex calculus problems.

Negative Integers

Negative integers often create confusion during differentiation, especially when using the Power Rule. However, understanding that they are nothing more than exponents that signify division helps simplify the process. When dealing with a function like \( x^{-m} \), think of it as \( \frac{1}{x^m} \).
This amounts to:

• First, rewrite the negative exponent term as a fraction.
• Then differentiate using techniques like the Quotient Rule, if necessary.

This rule makes it clear that even negative exponents follow the fundamental principles of differentiation, allowing you to handle them just like positive exponents.

Power Rule for Differentiation

The Power Rule for Differentiation extends beyond simple positive integers and applies to negative integers and other real numbers as well. This comprehensive rule helps calculate derivatives across a wide range of functions, ensuring consistency in solutions.

• Multiplier the exponent by the base to get the first part of the derivative.
• Subtract one from the exponent.
• Thus, derivatives such as \( x^{-m} \) follow the rule and simplify to \(-mx^{-m-1} \).

This helps solidify the Power Rule's versatility, allowing you to differentiate more complex expressions. By familiarizing oneself with how this rule operates across different types of exponents, it becomes easier to tackle differentiate functions efficiently.