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### Derivative of the exponential function

The derivative of the natural exponential function is

so this function is its own derivative!

##### Explanation

We start with the exponential function

Substitution in the definition of the derivative gives

You can write this as

For *x* = 0 you get

This is a constant, as there is no *x* in it. The value only depends on *a*, the basis. So you can write

We want to know for which value of *a* the function *f* ′(0) = 1 is, because then *f* ′(*x*) = *a ^{x}*, and this function is its own derivative. We calculate therefor

Multiplication with *h* gives

Switching sides shows

Exponentiation gives

what you can write as

Take , and as *h *→ 0 this gives *n *→ ∞, and so

This value is called the number *e* - of exponent. For that number you get

and the natural exponential function is it's own derivative!