I am working on elementary differential geometry and don't understand the following point: Let $S$ be a surface and let $\phi(u,v)$ be a local parametrisation of $S$ at $p \in S$. Why we have than that $$dN_p(\frac{\partial \phi}{\partial u})=\frac{\partial (N \circ \phi)}{\partial u}$$ where $N$ denotes the Gauss map.
Many thanks for some help!