I apologize for the second question on the same day, but I really do need to understand it.
We've got a consumption function (it's from economics but the question is still mathematical) looking like:
$$C=C(Y-T(Y))$$
where $C$ is the consumption, $Y$ is the income and $T(Y)$ is the tax as the function of income $Y$ (as in real world the taxes we pay are some proportion of the money we've earned);
What we need here is to differentiate this function in respect to Y (e.g. find $\frac {dC}{dY}$);
When browsing through internet, I've actually found a solution for this but I want to figure out how it's actually done (as it's a number of problems utilizing the same logic, so if I understand this one, it should be easy to do the rest).
In that solution it's done the following way
$$\frac{dC}{dY} = \frac{dC}{d(Y-T(Y))} * \frac{d(Y-T(Y))}{dY} + \frac{dC}{d(Y-T(Y))} * \frac{d(Y-T(Y))}{dT} * \frac{dT}{dY} $$
(and then it goes along so that the result looks more elegant)
I wonder which rule is used to do this. It somehow resembles the multivariable version of the chain rule, but the function we've got here is quite different from classical examples (like $x=y^2z^3$ while $y$ and $z$ are both some functions). Is it really the chain rule? Or something else?
The multivariable chain rule for, for example, a function of the form $(t,y(t)) \mapsto z$ is:
$$\dfrac{\mathrm dz}{\mathrm dt} = \dfrac{\partial z}{\partial t} + \dfrac{\partial z}{\partial y}\dfrac{\mathrm dy}{\mathrm dt}$$
How far you have to go to cover all combinations of derivative of such-and-such with respect to so-and so depends on what the variables implicitly depend on. If you play it safe and go further, and you find you didn't need to, you should get the right answer anyway.
ProofWiki
Khan Academy
Wikipedia