12
$\begingroup$

I can't believe I couldn't find this information online, but could someone provide me a good proof of the multivariate chain rule ? \begin{align} \frac{df}{dt} = \frac{df}{dx}\frac{dx}{dt} + \frac{df}{dy}\frac{dy}{dt} \end{align}

I found multiple derivation of this results online using differentials and mean value theorem, but they don't look like rigorous to me. Somehow dividing the differential by $dt$ doesn't make it rigorous for my point of view...

This question comes in a more general context where I am trying to understand why deriving a composition is effectively a matrix product. So by understanding this formula, I am able to see why building matrix of derivatives is a good tool to compute derivatives by matrix multiplication.

Thanks !

$\endgroup$
5
  • 1
    $\begingroup$ Have you you considered going to ProofWiki? They have hundreds of proofs, including the one you're looking for: proofwiki.org/wiki/Chain_Rule_for_Real-Valued_Functions Also note that they provide the book references, so you can easily check them elsewhere. $\endgroup$
    – zickens
    Dec 8, 2015 at 20:10
  • $\begingroup$ I'm checking it out ! thanks for the link. $\endgroup$
    – user149705
    Dec 8, 2015 at 20:17
  • $\begingroup$ Unfortunately the proof in your link use the "Characterization of differentiability" which just define a differentiable function using deltas. This looks like a circular proof to me... to prove the chain rule they divide by deltas, and to define the differentiability of a function they multiply by deltas ... $\endgroup$
    – user149705
    Dec 8, 2015 at 20:31
  • $\begingroup$ check the references! $\endgroup$
    – zickens
    Dec 8, 2015 at 20:35
  • 2
    $\begingroup$ this video : youtu.be/7eZVshlT33Q?t=1860 from MIT 18.02 Multivariable Calculus, Fall 2007 might help $\endgroup$
    – charany1
    Apr 26, 2017 at 13:41

2 Answers 2

11
$\begingroup$

Presumably we are saying that $f$ is a function of $x$ and $y$ (i.e., $f(x, y)$), which are both functions of $t\ \ $ ($x(t)$ and $y(t)$). So what does it mean to write $df/dt$? This is really the derivative of another function $F$ defined by

$$F(t) = f(x(t), y(t)).$$

Define the function $g$ by $g(t) = (x(t), y(t))$ so that $F(t) = f(g(t)) = f \circ g(t)$.

Recall the multivariable chain rule.

Theorem (Multivariable Chain Rule). Suppose $g\colon \mathbf{R}^n \to \mathbf{R}^m$ is differentiable at $a \in \mathbf{R}^n$ and $f\colon \mathbf{R}^m \to \mathbf{R}^p$ is differentiable at $g(a) \in \mathbf{R}^m$. Then $f \circ g\colon \mathbf{R}^n \to \mathbf{R}^p$ is differentiable at $a$, and its derivative at this point is given by $$D_a(f \circ g) = D_{g(a)}(f) \ D_a(g).$$

You can find a proof of this in, e.g., Calculus on Manifolds (Spivak). Back to the problem at hand: how do we use the chain rule to prove that

$$\frac{df}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}?$$

Well, let's try writing this in terms of a "matrix" product,

$$\frac{df}{dt} = \begin{bmatrix}\dfrac{\partial f}{\partial x} & \dfrac{\partial f}{\partial y}\end{bmatrix}\begin{pmatrix}dx/dt\\dy/dt\end{pmatrix}.$$

But this is exactly what the chain rule states when applied to the function $F = f \circ g$. We have that

  • $D_a(f \circ g) = D_a(F) = \dfrac{dF}{dt}$ (evaluated at some point $a$)
  • $D_{g(a)}(f) = \begin{bmatrix}\dfrac{\partial f}{\partial x} & \dfrac{\partial f}{\partial y}\end{bmatrix}$ (each term evaluated at $g(a)$)
  • $D_a(g) = \displaystyle \begin{pmatrix}dx/dt\\dy/dt\end{pmatrix}$ (each term evaluated at $a$)

where we have assumed differentiability of the maps.

$\endgroup$
2
  • 1
    $\begingroup$ I actually found the proof in this very book yesterday and finally understood what is going on here. I was a little bit corrupted by high school and undergrade mathematics to be honest... Thank you very much for the effort of this answer though it will be helpful to other for sure ! $\endgroup$
    – user149705
    Dec 10, 2015 at 13:55
  • 6
    $\begingroup$ But the theorem you've quoted is much more powerful, and usually uses the $n=1$ case in its proof. I'm not convinced you're not making a circular argument here. $\endgroup$
    – Teepeemm
    Mar 23, 2017 at 20:59
2
$\begingroup$

I also found most of the answers in the net circular and assuming the multiplication of two vectors for multivariable derivatives without proving it explicitly.

So I went back to basics and used the definition of derivative which is $\frac{f(x+e) - f(e)}{e}$ when $e \rightarrow 0$, which means $f(x+e) \simeq f(x) + e\cdot \frac{df}{dx}$

When applied to $f(x, y)$, then $f(x+e_{1}, y+e_{2}) \simeq f(x, y+e_{2}) + e_{1} \cdot \frac{df}{dx}\\ \simeq f(x, y) + e_{1} \cdot \frac{df}{dx} + e_{2} \cdot \frac{df}{dy}.(1)$

Also $x = x(t)$ and $y = y(t)$. So, $x(t+e) \simeq x(t) + e \cdot \frac{dx}{dt}\\ y(t+e) \simeq y(t) + e \cdot \frac{dy}{dt} (2)$

(by the way, here you can define a vector $z =[x, y]$ and assume by convention if $z$, $x$ and $y$ are functions of $t$ that $z(t+e) = [x(t+e), y(t+e)] \simeq z(t) + e \cdot [dx/dt, dy/dt]$. Here the array things start coming).

Finally, $f(t+e) = f(x(t+e), y(t+e))$, and as per (2)

$f(t+e) \simeq f(x(t) + e\frac{dx}{dt}, y(t) + e\frac{dy}{dt})$ and as per (1) $f(t+e) \simeq f(x(t), y(t)) + e\cdot\frac{dx}{dt}\frac{df}{dx} + e\frac{dy}{dt}\cdot\frac{df}{dx} = f(x, y) + e(\frac{dx}{dt}\cdot\frac{df}{dx} + \frac{dy}{dt}\cdot\frac{df}{dy}),$

But also $f(t+e) \simeq f(t) + e\cdot\frac{df}{dt}$. So when $e \rightarrow 0$, we got $\frac{df}{dt} = \frac{df}{dx}\cdot\frac{dx}{dt} + \frac{df}{dy}\cdot\frac{dy}{dt}.$ Not very rigorous proof though.

$\endgroup$
2
  • 3
    $\begingroup$ Welcome to MSE. Please use MathJax. $\endgroup$ Sep 6, 2017 at 9:54
  • $\begingroup$ The symbol $\simeq$ is not defined and this is not rigorous. The chain rule can be very easily intuited or roughly derived but this should have a proper proof $\endgroup$
    – FShrike
    Aug 6, 2023 at 22:22

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .