I struggle to break up things like $\frac {df}{dz}$ into partial derivatives as I'm not sure how it works.
I am reading and I see that if $f$ is a complex function and $z=x+iy$ that
$$ \frac {df}{dz}= \frac {\partial u}{\partial x}+i \frac{\partial v}{\partial x} $$
How does the LHS equal the RHS?
I am aware of the Cauchy Riemann equations if that is relevant.
I know it's an easy concept it's just it's always been brushed over and never properly taught to me.
Thanks
You can write $f(z)=f(x+i\, y)=u(x,y)+i\, v(x,y)$.
Now the whole idea behind the complex derivative is that it doesn't matter from which direction at a certain point in the complex plane you take the limit for the derivative.
If it would you would have infinite different derivatives in one point $z_0$ which isn't really useful at all.
So what follows is that $$\frac{\text{d}f}{\text{d}z}=\frac{\text{d}f}{\text{d}x}=\frac{1}{i}\frac{\text{d}f}{\text{d}y}$$ Ultimately we get $$\frac{\text{d}f}{\text{d}z}=\frac{\text{d}f}{\text{d}x}=\frac{\partial u}{\partial x}+i\,\frac{\partial v}{\partial x}$$