Let $n\geq 3$. The stereographic projection from $\mathbb{S}^n \backslash\{N\}$ to $\mathbb{R}^n$ is the inverse of
$$F: \mathbb{R}^n \rightarrow \mathbb{S}^n \backslash\{N\}, \quad x \mapsto\left(\frac{2 x}{1+|x|^2}, \frac{|x|^2-1}{|x|^2+1}\right)$$where $N$ is the north pole of $\mathbb{S}^n$. Define, for any $\lambda>0$,$$\psi_\lambda: x \mapsto \lambda x, \quad \forall x \in \mathbb{R}^n$$Set $\varphi=F \circ \psi_{\lambda} \circ F^{-1}$. My question is what's the value of $\left|\operatorname{det} \mathrm{d} \varphi(F(x))\right|$?