I have attached a scan because I don't know what the first line has to do with the rest. Has the printed a mistake?
Good eye -- for f(x) = x^(2x) you can not use the power rule as suggested by what follows
Power rule d (u^n)/dx = nu^(n-1)du/dx if n is a real number
however for x^(2x)
write y = x^(2x)
lny = 2xln(x)
y '/ y = 2ln(x) + 2
y ' = x^(2x)[2ln(x) + 2] but most certainly is not 2x*x^(2x-1)