In the following example from Chain Rule of Chapter 2 in Spivak's Calculus on Manifolds, I am confused about the dot after the first equality sign.
According to the chain rule, $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$. How is the circle changed to the dot? How to interpret this? Thanks in advance.
The function $f$ is equal to $\sin\circ g$, with $g(x,y)=xy^2$. Therefore,$$f'(x,y)=\sin'\bigl(g(x,y)\bigr)\circ g'(x,y).$$But $\sin$ is a function from $\Bbb R$ into $\Bbb R$. So, the linear map $\sin'(x)\colon\Bbb R\longrightarrow\Bbb R$ is simply the multiplication by the real number $\sin'(x)\bigl(=\cos(x)\bigr)$. So, it's a dot that since it is simply a multiplication.