Why is holomorphic function with non-zero derivative a conformal map?

Question

I am new to complex analysis, interested to know why non-zero derivative implies a conformal map.

Intuitively, I would think that non-zero derivative means the function is non-constant. Why would that be related to preserving angles?

Any intuitive reasons?

I understand that this may be a standard result in complex analysis. If so, please point out a good source where I can read more about it. (Suitable text for undergraduate level student)

Thanks!

Answer 1

If the derivative is nonzero then the complex derivative at a point is simply given by $d_pf(h) = ch$. In other words, the complex derivative is given by multiplication by a complex number. Any complex number which is nonzero can be written in polar form $c=re^{i \theta}$ hence the action of multiplying by $c$ is just a rescaling (dilation) and rotation of $h$.