Can we take the partial derivative of a function $f(x, y, z)$ with respect to $g(x, y, z)$

Asked Jan 01 '23 at 08:26

Active Jan 01 '23 at 13:20

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In Calculus 1 we differentiated $f(x)$ with respect to $g(x)$ that is, $\frac{df(x)}{dg(x)}$ by using chain rule: $$\frac{df(x)}{dx}\cdot\frac{dx}{dg(x)}= \frac{f'(x)}{g'(x)}$$

Is there something like this in partial derivatives too? For example: $$\frac{\partial f(x,y,z)}{\partial g(x,y,z)}$$ where $f(x, y, z)=xyz$ and $g(x, y, z)=x+y+z$

edited Jan 01 '23 at 13:20

joriki

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asked Jan 01 '23 at 08:26

Tobey

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There is a chain rule for the multivariable case. – Zag Jan 01 '23 at 09:14
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Does this answer your question? Differentiation of one function with respect to another in multivariable calculus? – Hans Lundmark Jan 01 '23 at 11:26
Consider a function $f(z, y)=z^2y$ where $z=xy$. Thus, $$\frac{\partial f}{\partial z}=2zy=2xy^2$$ But it is equivalent to saying:$$ \frac{\partial (x^2y^3)}{\partial (xy)}= 2xy^2$$ I'm puzzled here. – Tobey Jan 06 '23 at 09:24

Can we take the partial derivative of a function $f(x, y, z)$ with respect to $g(x, y, z)$

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