Chain rule (hopefully simple question?)

Question

I am trying to find the following partial derivative.

$$\frac {\partial }{\partial Y} U(Y-T(Y))$$

I know I need to use the chain rule, and if the function were simply U(T(Y)) it would obviously be straightforward - but I'm getting caught up on subtracting a function of Y from Y directly. Does anyone have any guidance on where to start here? (Also, I'm new to MathJax so let me know if I should make edits to make that better!)

Thanks all.

Answer 1

We have that

$$U=U(Y-T(Y))$$

then

$$\frac {\partial }{\partial Y} U(Y-T(Y))=U'(Y-T(Y))\cdot \frac {\partial }{\partial Y}(Y-T(Y))=U'(Y-T(Y))\cdot(1-T'(Y))$$