Let $$\mu(z)=\frac{az+b}{cz+d}$$ be a Möbius map. Then one can show that $\mu$ is holomorphic on $\mathbb{C}\backslash\{\frac{-d}{c}\}$ since $\mu$ can be decomposed into four holomorphic functions. But if we consider $\mathbb{C}_\infty$ and use the definition of the derivative at $\frac{-d}{c}$ then we will see that $\mu'(-d/c)=\infty$.
Also, if $c=0$, then $\mu$ is obviously holomorphic on $\mathbb{C}$. But, again, if we check $\mu'(\infty)$ using the definition, we will see that it is $\frac{a}{d}$.
Why is this incorrect?