I read the theory over and over again, but i don't get it.

You want the derivative of x^x. The theory says

"If , then . The chain rule is then used, to differentiate ln(y) with respect to x (y'/y). The product rule is then used on the right. Then solve for y'"

So you'd get

ln (y)'--> (1/y) =ln(x)+1<-- xln(x)'

Thus y= 1/ln(x)+1, which is wrong. The derivative of x^x should be:x^x(ln(x)+1)

1) Where is the mistake?

2)as the graph of y=f(x) is the same as the graph of ln(y)=ln(f(x)), why can you not simply derive the "right side", why do you have to derive both sides? In "normal" deriviation, you also don't derive the y!