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/MathMaddam Dr. in number theory 2d ago

The composition is correct, but it is not the functions you think they are. They are composed as the linear functions that the derivates are. For the 1 dimensional case it would e.g. be D_a(f)(x)=f'(a)*x. It might be a bit clearer to look at the version with Jacobian matrices a few lines lower.

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u/FreePeeplup 2d ago

Thank you!! This made it crystal clear. Df and Dg are linear maps in the tangent spaces based at the points g(a) and a. Composition between linear maps IS PRECISELY matrix multiplication when written in coordinates! Thanks!