Derivative of $x^n$.

We have proven that

• <Sign in to see all the formulas> when $n$ is a positive integer,
• <Sign in to see all the formulas> when $n$ is a negative integer and <Sign in to see all the formulas>
• and that <Sign in to see all the formulas> when n=0 because $x^0$ is a constant.

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>.