Working with exponents We start out with “base” number a. This number a must be positive, and we’re going to assume a > 1 to make it easier to draw graphs. What is the derivative of ax ? We’ll start to answer this question by reviewing what we know about exponents. To begin with, we know that: a0 = 1; a1 = a; a2 = a · a; a3 = a · a · a ... In general, ax1 +x2 = ax1 ax2 Together with the ﬁrst two properties, this describes the exponential function ax . From these properties, we can derive: x2 (ax1 ) = ax1 x2 and we can easily evaluate an for any positive integer n. For negative integers, we can see from the fact that am · a−m = am−m = 1 that a−m = a1m . We want to be able to evaluate ax for any number x; not just for integers. We start by deﬁning ax for rational values of x: √ p a q = q ap (where p and q are integers.) √ √ Since a1/2 · a1/2 = a1 = a · a, this seems like a reasonable deﬁnition. All that’s left is to deﬁne ax for irrational numbers; we do this by “ﬁlling in” the gaps in the function to make it √ continuous. This is what your calculator does when you ask it for the value of 3 2 or 2π . It can’t give you an exact answer, so it gives you a decimal (rational) number very close to the exact answer. Take some time and sketch the graph of 2x to “get a feel” for how exponential functions work.