Derivatives of Exponentials and Logarithms with Other Bases

The derivative of e^x is just e^x, no extra factor needed, because e is defined so that its own growth rate equals its own value. Any other base doesn't have that luxury. When you differentiate 2^x, a stray ln(2) shows up because 2 is not e, and the chain rule charges a toll for switching bases.

Here's the mechanism. Rewrite a^x as e^{x ln a}, using the identity a = e^{ln a}. Now differentiate the right side with the chain rule: the derivative of e^stuff is e^stuff times the derivative of stuff. Since stuff = x ln(a) and ln(a) is just a constant, its derivative is ln(a). So you get e^{x ln a} times ln(a), which is just a^x ln(a) once you swap back. The same trick handles logs: log base a of x equals ln(x)/ln(a), and since 1/ln(a) is a constant multiplier, the derivative is simply 1/(x ln a).

This topic is really the natural exponential and natural log rules wearing a disguise. Every other base reduces to e and ln once you rewrite it, so nothing new is being invented here, only chain rule and algebra applied to a substitution. That's why this sits right after the rules for e^x and ln(x) in this unit: those two are the foundation, and bases other than e are just chain-rule exercises built on top of them. Keep that connection in mind and this topic stops feeling like a new formula to memorize.