There are two ways to derive the number $e$. The way it was originally found, was thinking of compound interest. Let's say the interest rate is $z$ per year. If the interest is compounded once at the end of the year, you end up with $y=(1+z)$ dollars. If it is compounded at $x$ equidistant intervals, the money you'll have:
$$y = \left(1+\frac{z}{x}\right)^x$$
Taking the limit as $x \to \infty$ we get $y=e^z$
Alternately, we can look for a function that is the derivative of itself.
$$\frac{\partial y}{\partial z} = y$$
Solving this differential equation once again yields $y=e^z$.
I can't wrap my head around the intuitive connection between these two. Why should the function from compounding in the limit also have the property that it is its own derivative?
Compound interest means the rate your balance grows is proportional to how much you have. I.e. itd derivative is not necessarily itself, but it is a scalar multiple of itself. And because of how derivatives work with stetching fucntions, this means it can be stretched in a way to make a function which is a derivative of itself.
The more intervals you take, the closer your discrete compound gets to this continuous compound interest, i.e. if you join the dots, the more intervals you add, the closer the gradient of those lines is to actually being the scalar multiple of the current balance.