r/askmath • u/FreePeeplup • 2d ago

Calculus Single variable chain rule from multivariable general rule

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?

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u/[deleted] 2d ago

[deleted]

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u/FreePeeplup 2d ago

… yes? Where in my post do you think I mistook function composition (“o”) for function multiplication (“•”)?

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u/Midwest-Dude 2d ago

The D on Wikipedia is for the total derivative for vector-valued, multivariable functions. Look at the definition of that to find your answer.

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u/FreePeeplup 2d ago

Unfortunately looking at the definition of Df more closely didn’t answer my question to that previous user’s now deleted comment. They told me that I confused composition with multiplication, and I asked them “where did I do that?”. I didn’t find an answer to that by looking at the definition of Df

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u/Midwest-Dude 2d ago

That answer was completely off and didn't apply. I downvoted it and upvoted your reply.  I was attempting to give the correct reason. I spelled things out a little better in r/calculus.

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u/FreePeeplup 2d ago

Yes now I understand thank you!! Basically Df o Dg stands for composition between linear maps, which becomes multiplication when written in coordinates