Chain Rule for Real-Valued Functions/Corollary

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Theorem

Let $\Psi$ represent a differentiable function of $x$ and $y$.

Let $y$ represent a differentiable function of $x$.

Then:

\(\ds \frac {\d \Psi} {\d x}\) \(=\) \(\ds \frac {\partial \Psi} {\partial x} + \frac {\partial \Psi} {\partial y} \frac {\d y} {\d x}\)


Proof

\(\ds \frac {\d \Psi} {\d x}\) \(=\) \(\ds \frac {\partial \Psi} {\partial x} \frac {\d x} {\d x} + \frac {\partial \Psi} {\partial y} \frac {\d y} {\d x}\) Chain Rule for Real-Valued Functions
\(\ds \) \(=\) \(\ds \frac {\partial \Psi} {\partial x} + \frac {\partial \Psi} {\partial y} \frac {\d y} {\d x}\) Derivative of Identity Function

$\blacksquare$


Also see


Sources


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