 Solution

Let two regular differentiable paths through $z_0$ have nonzero tangent vectors $v_1,v_2$. The chain rule gives tangent vectors $f^{\prime }(z_0)v_1$ and $f^{\prime }(z_0)v_2$ for their images. Multiplication by a nonzero complex number is a positive scaling and rotation: writing $f^{\prime }(z_0)=r{e}^{i\theta }$ scales lengths by $r>0$ and adds the same angle $\theta$ to both tangents. Equivalently,Thus the oriented angle between the paths is preserved, proving $f$ is conformal at $z_0$. The nonzero derivative also gives local invertibility; a zero derivative could multiply angles through branching instead.