Derivative of $x^n$ where $n$ is a negative integer.

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Lesson Parent: 
Differentiation IV: Differentiation Formulas Everyone Must Know
Problem: 

Derivative of $x^n$ where $n$ is a negative integer.

Answer: 

$nx^{n-1}$ when $x\neq 0$.

SAMPLE SOLUTION

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Solution: 

Let's define <Sign in to see all the formulas> as a negative integer, $m$ as a positive integer, and that $m=-n$. Recall that we have proven this formula also holds for positive integers. Therefore, we know that

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since $m$ is a positive integer.

We can use the quotient rule to show that

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Because <Sign in to see all the formulas> and <Sign in to see all the formulas> are not defined at $x=0$

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holds except at $x=0$. Finally, note that the quotient rule requires that <Sign in to see all the formulas>. Since we use

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only at <Sign in to see all the formulas> then the quotient rule's requirement that <Sign in to see all the formulas> is automatically satisfied.