How does calculus without Euler's number (e) look?

What was meant in the video is that if you do calculus with $2^x$ for example, you get that

$$ \begin{aligned} \frac{d}{dx} 2^x = (\ln 2)2^x, \quad \int2^x dx = \frac{1}{\ln 2}2^x + C \end{aligned}$$

And of course, if you didn't know about $e$ and logarithms, $\ln 2$ would just be some crazy constant you have to figure out somehow. So the formulas for differentiation and integration are "not very nice". Compare this with $e^x$:

$$ \begin{aligned} \frac{d}{dx} e^x = e^x, \quad \int e^x dx = e^x + C \end{aligned}$$

Suddenly there's nothing to remember, no crazy constants, and differentiation and integration are as simple as they could possibly be.

Finally, if no-one had discovered $e$, I think that once calculus had been invented people would find it pretty fast. Once you know what differentiation is, it's natural to ask "is there a function that is its own derivative?" and this question will naturally lead to the function $e^x$.

The usual method for not introducing $e$ early is to do everything else, including integral and the fundamental theorem of calculus. Next, define a new function for $x>0$ by $$ f(x) = \int_1^x \; \frac{1}{t} \; dt $$ Then the exponential function is the inverse function of $f(x),$ call that $\exp, $ then the constant becomes $\exp 1$

\begin{align} \frac d {dx} 2^x & = \lim_{\Delta x\to0} \frac{2^{x+\Delta x} - 2^x}{\Delta x} = \lim_{\Delta x\to 0} \left( 2^x \frac {2^{\Delta x} - 1}{\Delta x} \right) \\[10pt] & = 2^x \lim_{\Delta x\to0} \frac {2^{\Delta x} - 1}{\Delta x} \quad \text{This step is possible because $2^x$ is} \\ & \quad \text{ “constant” in the sense that it does not change as $\Delta x$ approaches $0.$} \\[10pt] & = \Big(2^x \times \text{a constant}\Big) \text{ where this time “constant” means} \\ & \qquad \text{ not changing as $x$ changes.} \end{align} Similarly $$ \frac d {dx} 3^x = \Big(3^x \times \text{a constant} \Big) $$ but it's a different constant.

Now we have the problem of ascertaining what these "constants" are.

And instead of $2$ or $3$ as the base, for which number as base would the "constant" be equal to $1\text{?}$

Answering that last question is how the number $e$ would be discovered if we didn't already know about it.

And once we've done that, the laws of exponents plus the chain rule would would tell us that the two "consants" mentioned above are $\log_e 2$ and $\log_e 3.$