Asymptotic for the height of the derivatives of a rational function

Question

Let $\phi=\frac{P(z)}{Q(z)}$ be a homogeneous rational function of degree $d\ge 2$ over $\overline{\mathbb{Q}}$. If $h$ is the absolute logarithmic height, it seems that for each $z\in \overline{\mathbb{Q}}$, there is a constant $k_z$ such that

$h\left(\frac{\mathrm{d}^T}{\mathrm{d}z^T}\phi(z)\right)\sim k_z T\log T$, as $T$ goes to infinity.

Does anyone know:

1. If the asymptotic is correct?

2. How to prove it?