Question

Power Rule for negative integers Suppose \(n\) is a negative integer and \(f(x)=x^{n} .\) Use the following steps to prove that \(f^{\prime}(a)=n a^{n-1},\) which means the Power Rule for positive integers extends to all integers. This result is proved in Section 3.4 by a different method. Assume \(m=-n,\) so that \(m \geq 0 .\) Use the definition $$f^{\prime}(a)=\lim _{x \rightarrow a} \frac{x^{n}-a^{n}}{x-a}=\lim _{x \rightarrow a} \frac{x^{-m}-a^{-m}}{x-a}$$ Simplify using the factoring rule (which is valid for \(n>0\) ) \(x^{n}-a^{n}=(x-a)\left(x^{n-1}+x^{n-2} a+\cdots+x a^{n-2}+a^{n-1}\right)\) until it is possible to take the limit.

Short answer

Question: Prove the Power Rule for negative integers, specifically when \(n\) is a negative integer and \(f(x) = x^n\). Answer: The proof involves rewriting the expression with positive exponents, multiplying and dividing by the conjugate, simplifying the numerator using the factoring rule, canceling common factors, taking the limit, and finally simplifying the expression to obtain the desired result: \(f'(a) = na^{n-1}\). This proves that the Power Rule extends to all integers, including negative integers.

Step-by-step solution

Rewrite the expression

Rewrite the expression involving negative exponents as positive exponents by replacing \(n\) with \(-m\) where \(m \geq 0\). We have: \(f'(a) = \lim_{x\rightarrow a}\frac{x^{-m}-a^{-m}}{x-a}\)

Multiply and divide by the conjugate

In order to simplify the expression, multiply and divide the expression by the conjugate of the numerator, which is \((x^m+a^m)\): $$f'(a) = \lim_{x\rightarrow a} \frac{(x^{-m}-a^{-m})(x^m+a^m)}{(x-a)(x^m+a^m)}$$

Simplify the numerator using the factoring rule

Now, use the given factoring rule to simplify the numerator. Substitute \(n\) by \(-m\) in the expression: $$x^{-m}-a^{-m} = (x-a)(x^{(-m)-1} + x^{(-m)-2}a + \cdots + xa^{(-m)-2} + a^{(-m) - 1})$$

Rewrite the expression with simplified numerator

Plug the simplified numerator from Step 3 back into the expression from Step 2: $$f'(a) = \lim_{x\rightarrow a} \frac{(x-a)(x^{(-m)-1} + x^{(-m)-2}a + \cdots + xa^{(-m)-2} + a^{(-m)-1})}{(x-a)(x^m+a^m)}$$

Cancel \((x-a)\)

Notice that there is a common factor of \((x-a)\) in both the numerator and denominator. We can cancel these out: $$f'(a) = \lim_{x\rightarrow a} \frac{x^{(-m)-1} + x^{(-m)-2}a + \cdots + xa^{(-m)-2} + a^{(-m)-1}}{x^m+a^m}$$

Take the limit

Take the limit of the expression as \(x\) approaches \(a\): $$f'(a) = \frac{a^{(-m)-1} + a^{(-m)-2}a + \cdots + aa^{(-m)-2} + a^{(-m)-1}}{a^m+a^m}$$

Simplify the expression

Simplify the final expression and rewrite it to obtain the desired result: $$f'(a) = na^{n-1}$$ Thus, we have proved that the Power Rule for positive integers extends to all integers, including negative integers.

Key concepts

Limits in Calculus

Limits are a fundamental concept in calculus which provide a way to describe the behavior of functions as inputs approach a particular value. They form the foundation for defining derivatives and integrals. For instance, the definition of a derivative is constructed around the concept of a limit.

When dealing with limits involving power functions with negative exponents, such as \( f'(a) = \lim_{x\rightarrow a} \frac{x^{-m} - a^{-m}}{x - a} \), the idea is to understand what happens to the function as \(x\) gets arbitrarily close to \(a\). In the context of our exercise, limits help us establish the derivative of functions with negative powers by allowing us to rigorously define the slope of the tangent line at any point \(a\). Accurate calculation of limits can require various techniques, including factoring and rationalization, to simplify expressions so the limit can be easily determined as \(x\) approaches \(a\).

Derivative of Power Functions

The derivative of a function represents the rate at which the function's value changes with respect to changes in its input value. For power functions of the form \( f(x) = x^n \) where \(n\) is an integer, the Power Rule is a quick method for finding the derivative. The rule states that if \( f(x) = x^n \) then \( f'(x) = nx^{n-1} \).

This rule applies to both positive and negative integers for \(n\), which is demonstrated in our textbook problem by converting a power function with a negative exponent into an equivalent expression where the exponent is positive. The process shows that power rules are not just a mnemonic for common derivatives but rather principles founded on the limit definition of a derivative. Once the limit is properly manipulated, taking the derivative becomes a straightforward application of the Power Rule.

Negative Exponent Properties

Negative exponents follow specific algebraic rules which can simplify the manipulation of expressions involving them. The property \( x^{-n} = \frac{1}{x^n} \) tells us that a term with a negative exponent is equivalent to the reciprocal of that term with a positive exponent. This property is particularly useful when calculating derivatives or limits of power functions with negative exponents, as seen in the textbook exercise.

Understanding this property is crucial when working through calculus problems since it allows us to transform expressions into a form that enables the application of other calculus rules, like the Power Rule or factoring rules. The exercise demonstrates how we first turned a function with a negative exponent into one with a positive exponent to employ the rest of our calculus toolbox effectively.

Calculus Factoring Rules

In calculus, factoring rules are used to simplify expressions, particularly when we are working on problems involving limits, derivatives, or integrals. These rules make it easier to cancel out terms or to take limits by revealing common factors. A well-known factoring rule used for expressions \( x^n - a^n \) where \(n > 0\) is that this difference can be factored as \( (x - a)(x^{n-1} + x^{n-2}a + \cdots + xa^{n-2} + a^{n-1}) \).

Although this rule is generally provided for positive exponents, when dealing with negative exponents—as shown in our exercise—we adapt the rule by using the properties of negative exponents. This adaptation plays a key role in simplifying complex limits and ascertaining the derivative of power functions with negative exponents. By applying factoring rules, we can break down complex algebraic structures into more manageable pieces, enabling us to take the limit and finish the proof.