Question

Just starting to learn derivatives...
learned that if f(x)=e^x, f'(x)=e^x also. Blew my mind, really. but I decided to test it - because it should also be true that f'(x)=x(e^(x-1)). I came up with the conclusion that either one of them is wrong, or my work is wrong. Care to check my work?
x(e^(x-1))=e^x
if x=1, e^0=e^1
I'm somewhat confused.

Answer 1

Thats wrong. The rule \[ \frac{d}{dx} x^n = n x^{n-1} \] does only apply if n DOES NOT DEPEND ON x!

Answer 2

You can verify that \[ \frac{d}{dx} e^x = e^x \] by using the definition of e^x which is \[ e^x = \sum^\infty_{n=0} \frac{x^n}{n!} \]

Answer 3

Differentiate that sum term by term,

Answer 4

So... that rule only works when x is the base?

Answer 5

Yes.

Answer 6

And the exponent needs to be a constant!