$M,N,L$ are $n_{1},n_{2},n_{3}$-differentiable manifolds respectively, $m\in M$, $f:M\rightarrow N$ and $g:N\rightarrow L$ are $C^{\infty}$, and let $J_{f}(m)$ denotes the Jacobian of $f$ at $m$. I could prove $(gof)_{*}|_{m}=g_{*}|_{f(m)}of_{*}|_{m}$ but I couldn't complete that $J_{gof}(m)=J_{g}(f(m))\times J_{f}(m)$.
My works is below:
$J_{gof}(m)=\big[\frac{\partial(z_{i}(gof))}{\partial x_{j}}|_{m}\big]_{n_{3}\times n_{1}}$
$J_{g}(f(m))\times J_{f}(m)=\big[\frac{\partial(z_{i}og)}{\partial y_{j}}|_{f(m)}\big]_{n_{3}\times n_{2}}\times\big[\frac{\partial(y_{i}of)}{\partial x_{j}}|_{m}\big]_{n_{2}\times n_{1}}$ $=\big[\sum_{k=1}^{n_{2}}(\frac{\partial(z_{i}og)}{\partial y_{k}}|_{f(m)}\cdot\frac{\partial(y_{k}of)}{\partial x_{j}}|_{m})\big]_{n_{3}\times n_{1}}$
$\frac{\partial(z_{i}og)}{\partial y_{i}}|_{f(m)}\cdot\frac{\partial(y_{i}of)}{\partial x_{i}}|_{m}=g_{*}|_{f(m)}(\frac{\partial z_{i}}{\partial y_{k}})\cdot f_{*}|_{m}(\frac{\partial y_{k}}{\partial x_{j}})$
I think perhaps I should show that $\sum_{k=1}^{n_{2}}\big(g_{*}|_{f(m)}(\frac{\partial z_{i}}{\partial y_{k}})\cdot f_{*}|_{m}(\frac{\partial y_{k}}{\partial x_{j}})\big)=(g_{*}|_{f(m)}of_{*}|_{m})(\frac{\partial z_{i}}{\partial x_{j}})$ that it is $(gof)_{*}|_{m}(\frac{\partial z_{i}}{\partial x_{j}})$ from the first part.