Question on (Brouwer) degree of smooth functions Given smooth maps $f:M\to N$ and $g:N\to W$ (between closed, compact, oriented, $n$-manifolds), I want to show that $\text{deg}(g\circ f)=\text{deg}(g)\text{deg}(f)$ (Brouwer degree). 
I know, given, a regular point $x\in M$, that $sign\ d(g\circ f)_x= (sign\ dg_{f(x)})(sign\ df_x)$, but I'm not sure how I can extend that fact, or if that is even the right direction. 
The Brouwer degree is defined as follows (Milnor):
Let $f:M\to N$ be a smooth map (between closed, compact, oriented n-manifolds) and $p\in M$ a regular point of $f$ such that $df_x:T_xM\to T_{f(x)}N$ is a linear isomorphism. Define $sign\ df_x$ to be $\pm 1$, depending on whether $df_x$ preserves or reverses orientation. Then, for any regular value $q\in N$ 
$\text{deg}(f,q)=\sum_{p\in f^{-1}(q)} sign\ df_x$. Since, in fact, $\text{deg}(f,q)=\text{deg}(f,r)$ for any two regular values $q,r\in N$, $\text{deg}(f,q)$ is just called the (Brouwer) degree of $f$. 
 A: Let $w\in W$ be a regular value for $g\circ f$.  Note that, in particular, if $n\in N$ with $g(n)=w$, then $g(n)$ is also a regular value of $f$.
Then \begin{align} \sum_{x\in (f\circ g)^{-1}(w)} \operatorname{sign} d(g\circ f)_x &= \sum_{x\in (f\circ g)^{-1}(w)} \operatorname{sign} dg_{f(x)} \cdot \operatorname{sign} df_x \\ &= \sum_{n\in g^{-1}(w)} \, \, \sum_{x\in f^{-1}(n)} \operatorname{sign} dg_n \cdot \operatorname{sign} df_x \\ &= \sum_{n\in g^{-1}(w)} \operatorname{sign} dg_n  \cdot \sum_{x\in f^{-1}(n)} \operatorname{sign} df_{x}  \\ &= \sum_{n\in g^{-1}(w)}\operatorname{sign}dg_n \cdot  \deg(f,n) \\ &= \deg(f) \cdot \sum_{n\in g^{-1}(w)} dg_n \\ &= \deg(f) \cdot \deg(g,w)\\ &=\deg(f) \cdot \deg(g).\end{align}
The first line is the fact you quoted in your post.  Moving to the second is just the set theoretic fact that $(f\circ g)^{-1}(w) = \bigcup_{n\in g^{-1}(w)}f^{-1}(n)$. The third line uses the fact that $x$ varying does not affect $\operatorname{sign}dg_n$.  The fourth is the definition of degree.  The fifth uses the fact that $\deg(f,n_1) = \deg(f, n_2)$ for any regular values $n_i$ of $f$. 
