Mmh, this is a bit sloppy. The derivative of a function f::a -> b is a function Df::a -> a -o b where the second funny arrow indicates a linear function. I.e. the derivative Df takes a point in the domain and returns a linear approximation of f (the jacobian) at that point. And it’s always the jacobian, it’s just that when f is R -> R we conflate the jacobian (a 1x1 matrix in this case) with the number inside of it.

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matheist 4 days ago | parent | next [-]

Sorry to actually your actually, but the derivative of a function f from a space A to a space B at the point a is a linear function Df_a from the tangent space of A at a to the tangent space of B at b = f(a).

When the spaces are Euclidean spaces then we conflate the tangent space with the space itself because they're identical.

By the way, this makes it easy to remember the chain rule formula in 1 dimension. There's only one logical thing it could be between spaces of arbitrary dimensions m, n, p: composition of linear transformations from T_a A to T_f(a) B to T_g(f(a)) C. Now let m = n = p = 1, and composition of linear transformations just becomes multiplication.

(Only half kidding)

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btilly 3 days ago | parent | next [-]

The distinction between the space A and the tangent space of A becomes visually clear if we consider a function whose domain is a sphere. The derivative is properly defined on the tangent plane, which only touches the sphere at a single point. However in the neighborhood of that point, the plane and sphere are very, very close together. But are inevitably pulled away by the curvature of the sphere.

Of course that picture is not formally correct. We formally define the tangent space without having to embed the manifold in Euclidean space. But that picture is a correct description of an embedding of both the sphere and the tangent space at a single point.

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3 days ago | parent | prev | next [-]

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ziofill 3 days ago | parent | prev | next [-]

Oh I appreciate you actualling my actually ^^ but isn’t this case a special case of the one I wrote? I.e. when an and b are manifolds and admit tangent bundles?

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beng-nl 3 days ago | parent | prev [-]

Why, I’m sure you could come up with a succinct explanation of a monad :-)

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ndriscoll 4 days ago | parent | prev [-]

A perhaps nicer way to look at things[0] is to hold onto your base points explicitly and say Df:: a -> (b, a -o b) = (f(p),A(p)) where f(p+v)≈f(p)+A(p)v. Then you retain the information you need to define composition Dg∘Df=D(g∘f)=(Dg._1∘Df._1, Dg(Df._1)_.2∘Df._2). i.e the chain rule.

[0] which I learned from this talk https://youtube.com/watch?v=17gfCTnw6uE

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esafak 3 days ago | parent | next [-]

It's deplorable that we can't write in Latex or something similar here in 2025, and have to resort to the gobbledygook above.

	▲	xigoi 3 days ago \| parent \| next [-]
		Ironic considering that the web was originally designed for sharing scientific articles.
	▲	HeckFeck 3 days ago \| parent \| prev [-]
		But at least we FINALLY have an AI Assistant in... WhatsApp??

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ziofill 4 days ago | parent | prev | next [-]

Yes! I love Conal Eliot’s work. The one you wrote is the compositional derivative which augments the regular derivative by also returning the function itself (otherwise composition won’t work well). For anyone interested look up “the simple essence of automatic differentiation”.

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dbacar 4 days ago | parent | prev [-]

I respect the time you spent to write such a post with all those limited input alternatives (bowes).

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ndriscoll 4 days ago | parent [-]

You can do ≈ by long holding = on Android/Gboard. The only way I know to get ∘ is to copy/paste it from a Unicode reference. Likewise with ⊸, which I was too lazy to look up and didn't know the name of, but now I know is MULTIMAP (U+22B8).

	▲	tomsmeding 4 days ago \| parent [-]
		It's also \multimap in TeX. The name never made sense to me because while I've seen it used for a variety of linear functions in math, I've never seen it used for a multimap, and indeed the math name in common use for it seems to be "lollipop".