Derivative of xn where n is a negative integer.

Primary tabs

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

Derivative of xn where n is a negative integer.

Answer: 

nxn−1 when x≠0.

SAMPLE SOLUTION

All of our problems have fully worked out, logical solutions. We connect every solution to its underlying lesson so you can understand the concepts needed to solve any problem and reinforce your learning by practicing with similar problems. All of the formulas are available to our subscribers, but here is a sample of the problem solution you will see when you log in:



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

<Sign in to see all the formulas>


since m is a positive integer.

We can use the quotient rule to show that

<Sign in to see all the formulas>

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

<Sign in to see all the formulas>


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

<Sign in to see all the formulas>


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.