Differentiable Exponential Functions
The exponential property is actually a rather stringent condition on a differentiable function . Let’s start by assuming that is a differentiable exponential function and see what happens.
We calculate the derivative as usual by taking the limit of the difference quotient
Then the exponential property says that our derivative is
So we have a tight relationship between the function and its own derivative. Let’s see what happens for the exponential function . Since it’s the functional inverse of we can use the chain rule to calculate
Showing that this function is its own derivative. That is, this is the exponential function with .
Since a general (differentiable) exponential function is a homomorphism from the additive group of reals to the multiplicative group of positive reals, we can follow it by the natural logarithm. This gives a differentiable homomorphism from the additive reals to themselves, which must be multiplication by some constant . That is: . How can we calculate this constant? Take derivatives!
So our constant is the derivative from before. Of course we could also write
And since is invertible this tells us that . That is, every differentiable exponential function comes from by taking some constant multiple of the input.
By the usual yoga of inverse functions we can then see that every differentiable logarithmic function (an inverse to some differentiable exponential function) is a constant multiple of the natural logarithm . That is, if satisfies the logarithmic property, then