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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) = ax, 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!