Why do you see e everywhere in calculus?

For the same reason you hear Jamie Dimon of JPMorgan Chase talk about "the hockey stick." The hockey stick is shape of an exponential growth rate, like growth in profits after a start-up takes off...

hockey stick revenue growth

which is often just wishful thinking...

hockey stick magic

Or the growth of the national debt or the concentration of CO2 in the atmosphere...

growth of many kinds

The growth picks up speed. By the time you see it, it might be too late. Too late to stop it, or too late to buy in.

More sudden growth puts the handle flat and points the blade upwards:

sudden hockey growth

Compare ex

e to the x

Don't mistake e for a variable.

e not a variable

a is a variable. It could be any base with an exponent t.

In ect c and t are variables, e is not.

e2 has no variable in it. It's a constant.

You hardly ever see any base but e in calculus. Base ten and base two and others are easily translated into base e, and base e arithmetic is much easier.

translate to base e

Every exponential function, whatever the base, has the hockey stick shape. It's the shape of growth that picks up speed, that builds on itself.

Every exponential function has a derivative very similar to itself (some multiple of itself). It's a rate of change, and the derivative is the rate of change of that rate of change.

The convenience of e? The derivative of ex is ex. Someone (Bernoulli) went looking for the highest possible rate of compound (hockey stick) interest growth and found 2.71828, meaning that with continuous compounding (and a 100% interest rate to simplify the example) a dollar would grow to almost $2.72 in a year.

Just when you think e is no more interesting than compound interest on your savings, it turns up in something like the secretary problem:

You interview candidates for your administrative assistant. You ask personnel to send you 100 resumes and make the appointments. It's clear who's best when you interview them, but here's the hitch: you can't interview them all and then decide. You have to decide yes or no on each candidate when you interview him, and you can't go back later to someone you turned down. What are your odds of picking the best candidate, and what method gives you those odds?

Answer: Turn down the first 1/e candidates no matter what. Then take the first candidate who is better than any of the candidates you've seen. This gives you a 37% chance of picking the best of the 100 candidates.

Why 37%?

It's 1/e.

And e is elvish magic again.