ax and the Deﬁnition of the Derivative d x Our goal is to calculate the derivative dx a . It’s going to take us a while. We start by writing down the deﬁnition of the derivative d x ax+Δx − ax a = lim Δx→0 dx Δx We can use the rule ax1 +x2 = ax1 ax2 to factor out ax : d x a dx = = = ax+Δx − ax Δx→0 Δx ax aΔx − ax lim Δx→0 Δx Δx a −1 lim ax Δx→0 Δx lim As we’re taking this limit, we’re holding a and x ﬁxed while Δx changes (ap proaches zero). This means that for the purposes of taking this limit, ax is a constant. We can therefore factor the constant multiple out of the limit to get: d x aΔx − 1 a = ax lim Δx→0 dx Δx We’ve made a good start at ﬁnding the derivative of ax ; let’s look at what d x a is ax times some we have so far. We can see from our calculations that dx multiple whose value we don’t yet know. Let’s call that multiple M (a): aΔx − 1 . Δx→0 Δx M (a) = lim Using this deﬁnition of M (a), we can say that d x dx a = M (a)ax .

a^x and the Definition of the Derivative

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