Suppose we have two functions $f: \mathbb{R}^2 \to \mathbb{R}^2$ and $g: \mathbb{R}^2 \to \mathbb{R}^2$ so that $f$ and $g$ are inverses. So we have $f \circ g = g \circ f=I_2$.
My textbook states that from this and the chain rule it follows that $df \circ dg = dg \circ df = I_2$ as well.
The fact that we're using differentials confuses me. Could somebody show me how the above follows from the chain rule?
The derivative of the identity is again the identity. If $f\circ g=I$, the chain rule then tells you that $(f'\circ g)\,g'=I$.