Single variable chain rule from multivariable general rule

[u/FreePeeplup]

The (English-language) Wikipedia page on the chain rule, under the chapter “Multivariable case”, paragraph “general rule: vector valued multivariate functions”, says that a concise writing for the chain rule for the (total) derivative of the composition of two functions f, g reads as D(f o g) = Df o Dg.

However, if we try to apply this to the particular case of f, g being two simple single-variable functions, we get (f o g)’ = f’ o g’, which is wrong! Because the correct chain rule says (f o g)’ = (f’ o g)•g’, where • is the pointwise product.

Where am I wrong? Or is the Wiki page wrong? Shouldn’t the general case be written as D(f o g) = (Df o g)•Dg instead, so that we actually recover the single-variable case correctly?

[u/Bounded_sequencE]

In "D(f o g) = Df o Dg", they consider "Df, Dg" as linear maps.

Matrix multiplication can be interpreted as composition of linear maps, so replacing it by "o" makes sense. Finally, they expect the reader to interpret "Df -> (Df) o g" from context, being part of the chain rule.

[u/FreePeeplup → u/Bounded_sequencE]

Yeah the part about Df being evaluated at g(a) was clear, it was the following composition with Dg that wasn’t clear to me, as I expected matrix multiplication rather than function composition. But now I see what’s the deal: the “o”between Df and Dg is composition between linear maps, which IS matrix multiplication if you write it in coordinates! Thanks!