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I am a Maths major student.

Question: Given a function $f:\mathbb{R}^2\to\mathbb{R},$ $g:\mathbb{R}^2\to\mathbb{R}^2$ and $(a_1,a_2)\in \mathbb{N}^2.$ Assume that $f$ and $g$ are infinitely differentiable. Is there a formula for $$\frac{\partial^{a_1+a_2}} {\partial x_1^{a_1} \partial x_2^{a_2}} f(g(x_1,x_2))?$$

We need to use chain rule for multivariable version. I think it has something to do with tensor, as suggested by this post. But I am unable to pintpoint some references (books or lecture notes) to build my knowledge on tensor.

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The Wikipedia page on the chain rule has some info on this.

It will help to write $g=(g_1,g_2)$. Then, we have $$\partial_1(f(g_1(x_1,x_2),g_2(x_1,x_2))) = \partial_1f(g(x))\partial_1g_1(x) + \partial_2f(g(x))\partial_1g_2(x) = \partial_jf(g(x))\partial_1g_j(x).$$ Then, you just iterate this process and determine the general structure (find a pattern). Admittedly, these multivariable chain rule problems can get quite messy when computing higher derivatives.

Edit: To extend this to higher derivatives gets very involved beyond the second or third derivatives. You can look up Faà di Bruno's formula (there is a Wikipedia page on it) to see the generalized expression. I'm virtually sure you can find a proof online, so I won't reproduce it here. Unless you really can't find one, in which case I'll have a cup of coffee and give it a whirl.

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  • $\begingroup$ I know how to compute lower derivative. Question is that I am not able to see a pattern if I have higher derivative. $\endgroup$
    – Idonknow
    Mar 15, 2019 at 12:39

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