How to do Partial derivatives with respect to a function

Question

I am trapped by the question:

$f,g,h$ are functions of two independent variable $x,y$. Prove that:

$$\bigg(\frac{\partial f}{\partial g}\bigg)_h = \frac{1}{\bigg(\frac{\partial g}{\partial f}\bigg)_h}$$

and

$$\bigg(\frac{\partial f}{\partial g}\bigg)_x = \frac{\frac{\partial f}{\partial y}}{\frac{\partial g}{\partial y}}$$

( $\bigg(\frac{\partial f}{\partial g}\bigg)_h$ means doing partial derivative while treating $h$ as constant.)

There is not relationship between $f,g,h$, I can not do

$df = \frac{\partial f}{\partial g}dg + \frac{\partial f}{\partial h} dh$

since I cannot know if $f$ is a function of $g$ or $h$. Then I find no way to calculate $\frac{\partial f}{\partial g}$. Can anyone help?

Note: the context is physics problem so I'm sure $f,g,h$ are "good function" (continuous, differentiable, etc).

I was once warned not to treat partial derivative like total derivative. Can we really inverse $\frac{\partial f}{\partial y}$ into inverse $\frac{\partial y}{\partial f}$? @ — taper, Mar 08 '15 at 08:39
Is $\bigg(\frac{\partial g}{\partial h}\bigg)_h$ in the denominator intended as typed? Or should the subscript be $f$? — White Shirt, Mar 08 '15 at 08:46

score 0 · Accepted Answer · edited Apr 13 '17 at 12:21

I feel that some relationship between the three functions has to be implied in order for things like $\frac{\partial f}{\partial g}$ to make sense. (Perhaps sharing the same independent variables is sufficient -- hopefully someone more familiar with the topic than me can address that.)

If that is true, then consider

$$\frac{\partial f}{\partial x} = \frac{\partial f}{\partial g} \frac{\partial g}{\partial x} + \frac{\partial f}{\partial h} \frac{\partial h}{\partial x}$$

and likewise, $\frac{\partial g}{\partial x} = \frac{\partial g}{\partial f} \frac{\partial f}{\partial x} + \frac{\partial g}{\partial h} \frac{\partial h}{\partial x}$. Then by holding $h$ constant (i.e. $\frac{\partial f}{\partial h}=0$), the result for the first equation follows.

For the second part, consider

$$\frac{\partial f}{\partial g} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial g} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial g}$$

Holding $x$ constant eliminates the first term on the right-hand side. Regarding "reciprocals" of derivatives, perhaps the inverse function theorem or this question can shed some light.

How to do Partial derivatives with respect to a function

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