It's time for us to turn our attention to powers so that we may confront some impressive differential laws. There are two kinds of functions that can be built as powers: base functions and exponential functions. A base function allows its base \( b \) to vary. These include familiar functions like the squaring function
\[
\begin{align}
p &= b^2
\end{align}
\]
and the cubing function
\[
\begin{align}
p &= b^3.
\end{align}
\]
We'll see that reciprocals, square roots, and cube roots can also be written as base functions. And we'll learn a single differential law for base functions that unifies many of the differential laws we've already studied!
An exponential function, in contrast, allows its exponent \( e \) to vary. People use different exponential functions for different purposes. Engineers often like base ten exponentials
\[
\begin{align}
p &= 10^e
\end{align}
\]
since these can be handy when estimating the size of decimal numbers. Computer scientists using binary numbers prefer to work with base two exponentials
\[
\begin{align}
p &= 2^e.
\end{align}
\]
It turns out that another base is best for Calculus.
\[
\begin{align}
p &= \naturalbase^e
\end{align}
\]
Mathematicians have discovered that the natural base, \( \naturalbase \approx 2.718 \), gives the simplest differential law. Moreover, the natural base turns up when we take the differential of any exponential, natural or otherwise!