I'm reading the book "Teichmüller Theory and Applications to Geometry, Topology and Dynamics" by Hubbard and I have the following problem.
I need to compute $$\frac{d}{ds}\frac{\overline{\partial}z+\mu(t+s)\overline{z}}{\partial z+\mu(t+s)\overline{z}},$$ in $s=0$ and where $\mu$ is a Beltrami form on a Riemann Surface $X$ and $t\in[0,1]$. The books says that the complex derivatives are computed with respect to analytic coordinates on $X_{t\mu}$, i.e., with respect to $w=z+t\mu\overline{z}$. Then $$z=\frac{w-t\mu\overline{w}}{1-t^2|\mu|^2},\quad \overline{z}=\frac{\overline{w}-t\overline{\mu}{w}}{1-t^2|\mu|^2},$$ so that $$z+\mu(t+s)\overline{z}=\frac{w(1-t(t+s)|\mu|^2)+s\mu\overline{w}}{1-t^2|\mu|^2}$$
and the ratio of the derivatives becomes $$\left(\frac{\mu}{1-t(t+s)|\mu|^2}\right)\frac{d\overline{w}}{dw}. $$
I understand all the calculations before the last step, but I think that the ratio of the derivatives will be $$\left(\frac{s\mu}{1-t(t+s)|\mu|^2}\right)\frac{d\overline{w}}{dw}. $$
What I'm missing?
Thanks in advance !
For future reference, this is equation 6.6.30 in Hubbard's book on Teichmueller theory Vol. 1.
You are correct, for this equation to be correct is should read:
$$\frac{d}{ds}\left(\frac{s\mu}{1-t(t+s)|\mu|^2}\right)\frac{d\overline{w}}{dw}\Big|_{s=0}=\frac{\mu}{1-t^2|\mu|^2}\frac{d\overline{w}}{dw}.$$
This is a typo, since Hubbard ends up obtaining the correct result on the right hand side of the equation.