A Course in Differential Geometry by Thierry Aubin. Quote:"Let $f$ be a continuous map of an open set $\Omega\subset\mathbb{R}^n$ into $\mathbb{R}^p$...(and $f$ was at least $C^1$)... Let $g$ be a $C^1$ function of $\theta \in \mathbb{R}^p$ into $\mathbb{R}^m$. Suppose $f(\Omega)\in \theta$, then $g\circ f$ is a $C^1$ function of $\Omega$ into $\mathbb{R}^m$ and $(g\circ f)'=g' \circ f'$. "
I kind of understand what does $(g\circ f)'=g' \circ f'$ mean in multidimensional matrix.
However, when I think about $g,f$ both being one dimension, i.e. scalar functions, then shouldn't $(g\circ f)'()=g'\circ f()\cdot f'()$ according to chain rule?
Why differential and derivative are not consistent even in 1 dimension?
Given a (differentiable) function, f(x), the derivative is $f'(x)= \frac{df}{dx}$ is the usual $\lim_{h\to 0}\frac{f(x+h)- f(x)}{h}$. That is NOT a fraction but because it is a limit of a fraction, it can be treated as one. To make use of that we take "dx" as simply a symbol (but in applications can be treated as a very small number). Then we define $dy= f'(x)dx$. "dx" and "dy" are the differentials.