$g: S^2 \to \mathbb{R}^3, (x,y,z) \to (x,xy,xz)$
(1) Let $p \in S^2$. Find the rank of $(dg)_{p}: T_{p}S^2 \to T_{g(p)}\mathbb{R}^3$.($T_{p}S^2$ is tangent space at $p$)
(2) Show that $g(S^{2})$ is not a manifold.
I know that the tangent space of $S^{2}$ at p is a vector orthogonal to p.
In order to calculate the tangent maps, define the extension $\tilde{g}:\mathbb{R}^3\to \mathbb{R}^3$ by $\tilde{g}(x,y,z)=(x,xy,xz)$. By calculating the partial derivatives you get
\begin{align*} D\tilde{g}(x,y,z)=\begin{pmatrix}1&0&0\\y&x&0\\z&0&x \end{pmatrix}. \end{align*}
For any $p\in S^2$, we have \begin{align*} D\tilde{g}(p)|_{T_pS^2}=(dg)_p \end{align*} and using this relation you can easily calculate the rank of the tangent maps (just distinguish the cases $x=0$ and $x\neq 0$).
For the second part I suggest looking at the point $(0,0,0)\in g(S^2)$.