Chain Rule for Partial Derivatives/Corollary 1

Theorem

Let $F = \map f {x, y}$ be a real-valued function from $\R^2$ to $\R$.

Let $x = \map X t$ and $y = \map Y t$ be real functions.

Then:

$F = \map F t$

and:

$\dfrac {\d F} {\d t} = \dfrac {\partial F} {\partial x} \dfrac {\d x} {\d t} + \dfrac {\partial F} {\partial y} \dfrac {\d Y} {\d t}$

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

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Sources

• 1969: J.C. Anderson, D.M. Hum, B.G. Neal and J.H. Whitelaw: Data and Formulae for Engineering Students (2nd ed.) ... (previous) ... (next): $4.$ Mathematics: $4.4$ Differential calculus: $\text {(v)}$ Partial differentiation: $\text {(b)}$