Derivatives of Exponential and Logarithm Functions

An exponential function like $2^x$ or $e^x$ is simply one where each step to the right multiplies the output by the same amount, which is how populations, savings and radioactive decay behave. The big fact here is $\frac{d}{dx}e^x = e^x$: $e^x$ is the function whose slope always equals its height (it grows exactly as fast as it is big). The other formulas are small variations: $a^x\ln a$ (any other base, scaled by the constant $\ln a$) and $\frac1x$ (the slope of $\ln x$, the natural log, which undoes $e^x$). It's a short list, and each piece has a reason behind it.

Why $e$ is special. Try the definition on $a^x$:

Notice the $a^x$ factors right out, and what's left, $\frac{a^h - 1}{h}$, doesn't depend on $x$ at all. As $h \to 0$ it heads to some constant, which is just the slope of $a^x$ at $0$. So every exponential has a derivative proportional to itself: $\frac{d}{dx}a^x = a^x\cdot(\text{slope at } 0)$. For $a = 2$ that constant is about $0.69$; for $a = 3$ it's about $1.10$. Somewhere between 2 and 3 is the base where it's exactly $1$, and that's $e \approx 2.71828$. For that base, the derivative is the function itself. (The constant for a general base turns out to be $\ln a$, which is where the formula comes from.)

Logarithms. $\ln x$ is the inverse of $e^x$, so its graph is the graph of $e^x$ flipped across the line $y = x$. Flipping swaps rise and run, which turns every slope into its reciprocal. At the point $(x, \ln x)$, the matching point on $e^x$ has height $x$, so slope $x$, and flipping it gives slope $\frac1x$. More generally, $\frac{d}{dx}\ln|x| = \frac1x$ for every $x \ne 0$.

Other bases come from converting to base $e$: $a^x = e^{x\ln a}$ and $\log_a x = \frac{\ln x}{\ln a}$.

Exponential growth and decay have rates proportional to their size, $y' = ky$, which is exactly why $e^{kt}$ solves them: populations, radioactive decay, continuously compounded interest.