Derivative of $x^n$.

Primary tabs

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

Determine the derivative of $x^n$ for any integer $n$.

Answer: 

$n\,x^{n-1}$

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: 

We have proven that

The three bullet points above account for every scenario except when both $n=0$ and $x=0$. For $n=0$ the formula <Sign in to see all the formulas> gives the same answer as the general formula

<Sign in to see all the formulas>


except when $x=0$. When $x=0$ you get division by zero and the formula <Sign in to see all the formulas> can't be used. However, it matters very little as we see from the actual definition of the derivative that

<Sign in to see all the formulas>


because <Sign in to see all the formulas>. This makes intuitive sense because $x^0$ is a constant and the derivative of a constant is zero. (Note that some math aficionados will say that $0^0$ can also equal zero. That's true, but beyond the scope of this problem. Just think of $0^0=1$ and move on.)

Therefore, you can use the formula <Sign in to see all the formulas> for all integers $n$ except at $x=0$. When $x$ does equal zero then the formula can be used for <Sign in to see all the formulas>.