The derivative of a Mobius transformation
by Hidenori
Proposition
Find the derivative of the function $T(z) = \frac{az + b}{cz + d}$ where $a, b, c, d \in \mathbb{C}$ with $ad - bc \ne 0$. When is $T’(z) = 0$?
Solution
\[\begin{align*} T'(z) &= \frac{(cz + d)a - (az + b)c}{(cz + d)^2} \\ &= \frac{ad - bc}{(cz + d)^2}. \end{align*}\]Therefore, $\forall z \in \mathbb{C}$, $T’(z) \ne 0$.
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