Let $G$ be a linearly reductive group acting regularly on an irreducible affine variety $X$ (over an algebraically closed field of characteristic zero). Suppose there's a $G$-stable open subvariety $U$ of $X$ such that the geometric quotient $\Phi: U\rightarrow W$ exists. Also, assume that $W$ is a quasi-projective variety of $\mathrm{dim}\,W\geq1$.
Let $x\in U$ be a point which is smooth in $X$. I want to know how are the tangent spaces $T_{x}(X), T_{x}(\mathcal{O}(x))$ and $T_{\Phi(x)}(W)$ related, and how are their dimensions related?
