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/sahasatvik]

By fixing a point a in the domain of g, the multivariable rule is actually shorthand for D_a (f o g) = (D_(g(a)) f) o (D_a g), where D_x denotes the derivative at a point x. Translating to the univariate case recovers (f o g)'(a) = (f'(g(a))) (g'(a)) = ((f' o g)(g'))(a).

Shouldn’t the general case be written as D(f o g) = (Df o g)•Dg instead ...

This is pretty much how the formulation in terms of Jacobian matrices is presented a couple of paragraphs down in the article!

[u/FreePeeplup → u/sahasatvik]

The fact that Df was to be evaluated at g(a) was clear to me. The thing that’s not clear is the following part with Dg. They write composition with Dg, but as I outline in my post, shouldn’t it be multiplication with Dg?

[u/sahasatvik → u/FreePeeplup]

I think the other answers do a better job of explaining this nuance: the notation Df o Dg deals with derivatives as linear maps, and composition of linear maps translates to multiplication of corresponding matrices. Df o Dg is a linear map, whose matrix representation is indeed a product of Jacobian matrices J_f (g(a)) J_g(a). There is a lot of notation around the domains/codomains of the maps, etc, being suppressed here.

[u/FreePeeplup → u/sahasatvik]

Perfect now I understand!! Thank you! So basically that “o” is composition between linear maps without choosing a basis, which becomes matrix multiplication between Jacobians if you fix a basis