In Ahlfors Complex Analysis, the author writes the equation
$$|f'(z)|^2=\left( \frac{\partial u}{\partial x}\right)^2+\left( \frac{\partial v}{\partial y}\right)^2$$
I am wondering why this is ture. I know that $f'(z)=\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}$ so if we were to take the modulus of $f'(z)$, wouldn't we instead get $$|f'(z)|=\left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial v}{\partial x}\right)^2$$
Since $|x+yi|=x^2+y^2$