You have the following function:
f(x) = xˣ
In order to differentiate the previous function, you proceed as follow:
apply ln to the function:
ln f(x) = ln (xˣ)
use the property of logarithm ln aⁿ = n(lna)
ln f(x) = x (ln x)
differentiate the previous equation
[ln f(x)]' = ln x + x(1/x) = ln x + 1
[ln f(x)]' = ln x + 1
use [ln f(x)]' = f'(x)/f(x)
f'(x)/f(x) = ln x + 1
f'(x) = f(x)[ln x + 1]
replace f(x) = xˣ
f'(x) = xˣ[ln x + 1]