Differential of restriction vs. restriction of differential
Let $\jmath:N\hookrightarrow M$ be the inclusion map. Then:
$$\phi|_N=\phi\circ \jmath.$$
Given $p\in N$, it follows
$$d(\phi|_N)_p=d(\phi\circ \jmath)_p=d\phi_{\jmath(p)}\circ d\jmath_p=d\phi_p\circ d\jmath_p$$
Notice that $T_pN\subset T_pM$ via the injection (recall $\jmath$ is an immersion):
$$d\jmath_p: T_pN\rightarrow T_{\jmath(p)}M=T_{p}M.$$
Hence,
$$d\phi_p|_{T_pN}=d\phi_{p}\circ d\jmath_p=d(\phi\circ \jmath)_p=d(\phi|_N)_p.$$
So the answer is yes!