I try to compute the Fubini-Study metric $h_{FS}$ in an affine chart. To this end, let $\phi: \mathbb{C}^n \to S^{2n+1}$ given by $\phi(z_1, ..., z_n) := \frac{1}{\sqrt{1+|z|^2}} (1, z_1, ..., z_n)$, where $z=(z_1, ...., z_n)$.
Let $\mathbf{z} = (z_0, z)$ be the coordinate of $\mathbb{C}^{n+1}$. Since $h_{FS}$ is the pullback of $d\mathbf{z} d \mathbf{z}$ under $\phi$, we need to compute the components of $d\phi \overline{d\phi}$.
For $i \geq 1$, I have computed that $$ (d\phi)_{ik} = \frac{\partial \phi^i}{\partial z_k} = \frac{\delta_{ik}}{\sqrt{1+|z|^2}}- \frac{z_i \overline{z_k}}{2(1+|z|^2)^{3/2}}, $$
but then $$ d\phi \overline{d \phi} \neq \frac{(1 + |z|^2)\delta_{ij} - \overline{z_i}z_j}{(1+|z|^2)^2}. $$
Is the expressionf or $d\phi$ correct? Can somebody help me finalize the computation?
P.S. I am aware of the following question: Computing the Fubini-Study metric, but it is only outlining the strategy.