Different ways to express partial derivatives

Question

I have a teacher of economics who likes to use mathematical proofs all the time. Specifically in one exercise he proposed to use the partial derivative of the total expenditure of a economy with respect to the interest rate to show that it is a decreasing function. I explain. The function of the total income / expenditure in a closed economy is defined as follows (is-lm model): $ Y = C(R , Y - T) + I(R) + G$. Where Y,the total expenditure, is equal to the total consumption of the agents (which depends of the interest rate R and the disposable income, that is income minus taxes); the total investment made by firms (which also depends of the interest rate R) and finally the government expending G (which is exogenus to the model). Now here's the part i get confused, how can I derive a function I do not know what looks like? How should I interpret the $C(R, Y-T)$ part mathematically? I know that is a function that depends on several variables, but how could I represent generically the derivative it will yield? I've tried to look at general forms of chain rule and product rule and then apply it to the case but it failed. Since i did not see this proof nowhere in textbooks and over the internet I'm starting to consider that his answer maybe wrong. Here it goes:

Answer 1

Write $C=C(x,y)$ where $x=R$ and $y=Y(R)-T$. Then, we have

$$\begin{align} \frac{\partial C(x,y)}{\partial R}&=\frac{\partial C(x,y)}{\partial x}\frac{\partial x}{\partial R}+\frac{\partial C(x,y)}{\partial y}\frac{\partial y}{\partial R}\\\\ &=\left(\left.\frac{\partial C(x,y)}{\partial x}\right|_{x=R,y=Y(R)-T}\right)\frac{\partial R}{\partial R}+\left(\left.\frac{\partial C(x,y)}{\partial y}\right|_{x=R,y=Y(R)-T}\right)\frac{\partial Y(R)-T}{\partial R}\\\\ &=C_1(R,Y(R)-T)\times 1+C_2(R,Y(R)-T)\times \frac{\partial Y(R)}{\partial R}\\\\ &=C_1(R,Y(R)-T)+C_2(R,Y(R)-T) \frac{\partial Y(R)}{\partial R} \end{align}$$

where

$$C_1(R,Y(R)-T)\equiv\left.\frac{\partial C(x,y)}{\partial x}\right|_{x=R,y=Y(R)-T}$$

$$C_2(R,Y(R)-T)\equiv\left.\frac{\partial C(x,y)}{\partial y}\right|_{x=R,y=Y(R)-T}$$