Chain Rule for Partial Derivatives/Corollary 2

Theorem

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

Let $y = \map Y x$ be a real function.

Then:

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

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: $\text {(a)}$