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Non-math student here so go easy on me.

How do we calculate a partial derivative in terms of $x$ when dealing with a multivariable composite function, and what 'chain rule version', if any, could one refer do?

The function I have in mind is, $$f(x,g(x,y))$$ and I want $\frac{\partial f}{\partial x}$.

Mice Andealfe
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1 Answers1

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First we have to be careful with the notation. We will introduce variables $u$ and $v$ to denote the "first" and "second" arguments of $f$ so that we can write the function as $f(u,v)$.

Now you're interested in differentiating $f(x,g(x,y))$ with respect to $x$. Here we have $u=x$ and $v=g(x,y)$. The chain rules tells us to compute the following,

$$ \frac{\partial f}{\partial x} = \frac{\partial u}{\partial x} \left( \frac{\partial f(u,v)}{\partial u} \right)_{u=x,v=g(x,y)}+ \frac{\partial v}{\partial{x}} \left( \frac{\partial f(u,v)}{\partial v} \right)_{u=x,v=g(x,y)}$$

$$ = \left( \frac{\partial f(u,v)}{\partial u} \right)_{u=x,v=g(x,y)}+ \frac{\partial g(x,y)}{\partial{x}} \left( \frac{\partial f(u,v)}{\partial v} \right)_{u=x,v=g(x,y)}$$

Spencer
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