Why is there an intimate relationship between calculus and $e$?

Question

Why did the irrational number $e$ become such a crucial part of calculus? Why didn't some other irrational number like, say, $\pi$ take the place of $e$?

Answer 1

The roles of $e$ and $\pi$ are similar in calculus.

The number $e$ arises naturally in the solution of the simplest possible first order differential equation because $y(t)=e^t$ is the solution of $y'=y$.

In similar fashion, $\pi$ arises in the solution of the the simplest second order differential equation because $y(t)=\sin(t)$ and $y(t)=\cos(t)$ are the solutions of $y^{\prime\prime}=-y$.

While not immediately calculus related, it might also be mentioned that the golden ratio arises as the solution of one of the simplest algebraic equations, namely $x^2-x-1=0$.

Answer 2

$e$ is directly related to derivatives because the function $x\mapsto e^x$ is its own derivative. No other base has this property.

Meanwhile, $\pi$ is also intimately related to calculus, because any function whose second deriviative is the negative of the original function will be periodic with period $2\pi$.