why f(x)=e^x is it's own derivative?

because by the Chain Rule:

we have two functions f(x)=e^(x) and g(x)=x

hence f(g(x)) = e^(x)

f'(g(x)) is then by the Chain Rule f(g(x)) * g'(x) = e^(x) * 1 = e^(x)

the short version:

the derirative rule for all a^x expressions is:
f' a^x = a^x * ln (a)

plug in e for a:

f' e^x = e^x * ln (e) = e^x * 1 = e^x

Because this function has been invented precisely for this.

The derivative finds the slope of a function at any point right.
And e^x is a graph where the slope is always equal to the y value.
Therefore the derivative of the function itself is the same.

isn't that just circular reasoning?

Isn't that what the function and derivative of it is?

i didnt say you were wrong. i just suggested it was a circular argument. he asked why the derivative of e^x is e^x, and you basically responded the derivative of e^x is e^x, therefore the derivative of e^x is e^x

A real math question is : why does it only exist 1 function f (up to a multiplicative constant) such that f ' = f ?

A second question is : why is f called "exponential" ?

Or you could start with a different definition of exponential, and try to see why it is its own derivative. But in this case, you must start with an explicit definition.

This post was edited by feanur on Jun 20 2015 12:39am