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The Power Rule

Here is one of the most basic rules of calculus, perhaps the single rule that more people think of when someone says, ``What do you remember from calculus?" Besides, it has that great name, The Power Rule.

It says:

\begin{displaymath}\boxed{ \frac {d} {dx} (x^n) = n x^{n-1} }\end{displaymath}

Now what does that mean? Well, n can be any number. So if we let n be 3, we have:

\begin{displaymath}\frac {d} {dx} (x^3) = 3x^2.
\end{displaymath}

If we let n be 5, we have:

\begin{displaymath}\frac {d} {dx} (x^5) = 5x^4.
\end{displaymath}

If we let n be 1, we have:

\begin{displaymath}\frac {d} {dx} (x^1) = 1x^0=1.
\end{displaymath}

This is worth remembering separately, so let's repeat it so it sticks:

\begin{displaymath}\frac {d} {dx} (x) =1.
\end{displaymath}

We could also let n be a negative number. For instance, if n=-2, we have

\begin{displaymath}\frac {d} {dx} (x^{-2}) = -2x^{-3}.
\end{displaymath}

Notice that we can now take the derivative of 1/x:

\begin{displaymath}\frac {d} {dx} (\frac {1}{x}) =\frac {d} {dx} (x^{-1}) = -1x^{-2} = \frac {-1}{x^2}
\end{displaymath}

In fact, n can even be a fraction. So for instance, we can take the derivative of $\sqrt{x}$, since:


\begin{displaymath}\frac {d} {dx} (\sqrt{x}) =\frac {d} {dx}(x^{1/2}) = \frac{1}{2}x^{-1/2}
\end{displaymath}

The fact that n needn't be a whole number usually doesn't get mentioned in this part of a course on calculus, but we'll clue you in a little early. That little n could even be a number like $\pi$ or $\sqrt 2$. Then:

\begin{displaymath}\frac {d} {dx} (x^{\pi}) = {\pi}x^{\pi-1}
\end{displaymath}

We know it's tempting to try to simplify the exponent $\pi-1$, but it's best to leave it the way it is. Just think of it as a number with value about 2.1416.


Good Question: Is it possible to prove the power rule  for whole numbers using just induction and the product rule?


Of course, there is a little risk here. This question makes you sound exceptionally intelligent, and will momentarily lift you onto a godlike plateau miles above the rest of the class in the mind of the professor. If the professor swallows the bait, and uses induction and the product rule to prove the power rule, you're set for life, or at least for the rest of the semester. However, if the professor doesn't bite, but instead says, ``How do you mean?" and you respond,``Um, ah, oh I don't know, I just made it up," it could look very bad.



Next: The Product Rule Up: Derivatives: How to Find Previous: The Basic Rules of
Joel Hass
1999-05-26