Chain Rule for Partial Derivatives/Multivariable

Theorem

Let $\map f {x_1, x_2, \ldots, x_n}$ be a real-valued function of $n$ variables.

Let $x_1, x_2, \ldots, x_n$ be real-valued functions of $m$ variables $t_2, t_2, \ldots, t_m$.

Then for all $i \in \Z$ such that $1 \le i \le m$:

$\ds \dfrac {\partial f} {\partial t_i}$

$=$

$\ds \dfrac {\partial f} {\partial x_1} \dfrac {\partial x_1} {\partial t_i} + \dfrac {\partial f} {\partial x_2} \dfrac {\partial x_2} {\partial t_i} + \cdots + \dfrac {\partial f} {\partial x_n} \dfrac {\partial x_n} {\partial t_i}$

This article is complete as far as it goes, but it could do with expansion.
In particular: Conditions on continuity and/or differentiability need to be incorporated here
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Expand}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Proof

This theorem requires a proof.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): chain rule (multivariable)