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I have a function of $f(x_1, x_2)$, and two other functions of $g_1(x_1, x_2)$ and $g_2(x_1, x_2)$ in such a way that the original function can be written as $ f(g_1, g_2) $.

Now I need to fine partial derivative of $f$ with respect to $g_1$ or $g_2$ while keeping the other constant. That is

$$ \Big( \dfrac{\partial f}{\partial g_1} \Big)_{g_2 = const} $$

and

$$ \Big( \dfrac{\partial f}{\partial g_2} \Big)_{g_1 = const} $$

What is the correct formula to do this?

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  • $\begingroup$ What are the domains and codomains of your functions? $\endgroup$ – user258700 Jan 3 '16 at 23:00
  • $\begingroup$ Don't really know. I am more of a structural engineer trying to sort out some form of equation for simulation... I think your question goes way beyond my scope... $\endgroup$ – Quang Thinh Ha Jan 3 '16 at 23:03
  • $\begingroup$ Not really. I am just asking where do $f$ and $g$ get their inputs from, and what do they output. For instance, is $f:\Bbb R^2 \to \Bbb R$? $\endgroup$ – user258700 Jan 3 '16 at 23:05
  • $\begingroup$ Ah I see. Yes they do. $\endgroup$ – Quang Thinh Ha Jan 3 '16 at 23:07
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This is probably not the most rigorous or direct way of doing it, but it should explain what's going on:

By the chain rule, we can write $$ \begin{align} \mathrm{d}f &= f_x\ \mathrm{d}x + f_y\ \mathrm{d}y \\ \mathrm{d}g_1 &= g_{1x}\ \mathrm{d}x + g_{1y}\ \mathrm{d}y \\ \mathrm{d}g_2 &= g_{2x}\ \mathrm{d}x + g_{2y}\ \mathrm{d}y \\ \end{align} $$

Let us think about $(\partial f/\partial g_1)_{g_2}$. The condition $g_2 = \text{constant}$ means that we are confined to move along a contour of $g_2$, so let us first work out what those contours are, and then work out the rates of change of $f$ and $g_1$ if we move along such a contour.

Since $g_2 = \text{constant}$ along a contour, we must have $$ g_{2x}\ \mathrm{d}x + g_{2y}\ \mathrm{d}y = 0 $$ and so $$ \mathrm{d}y = - \frac{g_{2x}}{g_{2y}}\ \mathrm{d}x $$ on contours of $g_2$ (but see Note 1). The contours are therefore described by the differential equation $$ \frac{\mathrm{d}y}{\mathrm{d}x} = - \frac{g_{2x}}{g_{2y}}. $$

Now we have a relationship between $\mathrm{d}y$ and $\mathrm{d}x$. We shall substitute these into the expressions for $\mathrm{d}f$ and $\mathrm{d}g_1$ to work out the rates of change of $f$ and $g_1$ along the contours.

For $f$, we have $$ \mathrm{d}f = \left(f_x - \frac{g_{2x}}{g_{2y}} f_y \right)\ \mathrm{d}x $$ and a corresponding equation for $\mathrm{d}g_1$. Dividing the two (see Note 2) then gives the desired expression: $$ \left(\frac{\partial f}{\partial g_1}\right)_{g_2} = \frac{ f_x g_{2y} - f_y g_{2x} }{ g_{1x} g_{2y} - g_{1y} g_{2x} }, $$ or, in terms of 'Jacobian determinants': $$ \left(\frac{\partial f}{\partial g_1}\right)_{g_2} = \frac{ \partial(f,g_2)/\partial(x,y) }{ \partial(g_1,g_2)/\partial(x,y) }. $$

Notes

  1. If $g_{2y}=0$, then we can proceed instead by writing $\mathrm{d}x$ in terms of $\mathrm{d}y$ instead, and proceed accordingly to get the same result.

    If both $g_{2x}$ and $g_{2y}$ are zero at the same point, then $g_2$ is stationary there, and $(\partial f/\partial g_1)_{g_2}$ is not defined at that point.

  2. The partial derivative $(\partial f/\partial g_1)_{g_2}$ is undefined at points where $\partial(g_1,g_2)/\partial(x,y) = 0$. At such points, the gradients of $g_1$ and $g_2$ are not linearly dependent, which means that the contour of $g_2$ is also a contour of $g_1$; the rate of change of $g_1$ is therefore zero there.

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