3Blue1Brown — What's so special about Euler's number e?

Summary: Derives the rule for differentiating exponential functions and shows that what makes $e$ special is being the unique base whose exponential equals its own derivative.

Core argument

1. Setup. Treat $2^t$ as a continuously growing population mass. The “rate over a full day” equals the population at the start of the day, suggesting (incorrectly) that $\frac{d}{dt}{2}^t=2^t$.
2. Shrink the interval. The actual derivative is

$$\frac{d}{dt}{2}^t=\underset{dt\to 0}{\lim }\frac{2^{t+dt}-2^t}{dt}$$

The exponential identity $2^{t+dt}=2^t\cdot 2^{dt}$ lets us factor out $2^t$, leaving

$$\frac{d}{dt}{2}^t=2^t\cdot \underset{dt\to 0}{\lim }\frac{2^{dt}-1}{dt}$$

3. Mystery constant. The bracketed limit is a constant independent of $t$ — substituting a very small value for $dt$, numerically the constant $\approx 0.6931$ for base 2, $\approx 1.0986$ for base 3, $\approx 2.079$ for base 8. So every exponential is proportional (but, in almost all cases, not equal) to its own derivative.
4. Defining $e$, the special case. $e\approx 2.71828$ is the unique base where that proportionality constant is exactly 1.
   • For this value only, after exponentiating it as $e^t$, we note the following:
   • The derivative $\frac{d}{dt}{e}^t$, is not just proportional, but precisely equal to the exponential $e^t$ itself.
   • $\frac{d}{dt}{e}^t=e^t$ is, in a sense, what defines $e$.
5. The rewrite. Any $a^t$ can be written as $e^{ct}$ for some $c$. The mystery constant for base $a$ is $\ln (a)$, the answer to ”$e$ to the what equals $a$”:

$$\frac{d}{dt}{a}^t=a^t\cdot \ln (a)$$

6. Why we care. Many natural processes — population growth, Newton’s cooling, compound interest — have a rate of change proportional to the quantity itself. Writing solutions as $e^{ct}$ makes the proportionality constant $c$ literally visible in the exponent.

Key derivation steps

• Exponential identity: $a^{x+y}=a^x\cdot a^y$ — the property that turns additive inputs into multiplicative outputs and is the lever used in the derivative derivation.
• Chain rule rewrite: $a^t=e^{\ln (a)\cdot t}$, so $\frac{d}{dt}{a}^t=e^{\ln (a)t}\cdot \ln (a)=a^t\ln (a)$.

Sources

• What’s so special about Euler’s number e?