Product rule for 3 or more variables?

Question

I saw another question regarding the product rule for 3(or more) variables here: Finding derivative of three variables I used the second answer in one of my exam papers and got the final answer correct, but I lost a few marks because I didn't use the same method (I think). I've asked a few people and they say that my working out is fine. I don't really know what is wrong, my working out or other exam specific things. I don't want anybody else to lose marks on their exams because of this, so i just want to make sure its correct. I've added a picture of my working out (part a and marks on top right) and the exam mark scheme to this. Also, do you think I should get a remark for this exam? Any help would be much appreciated. I'm sorry if this question is bad, I just thought I would post it here since I'm not sure whether the other answer is wrong. Thanks.

Answer 1

You have differentiated the $y^{-1}$ incorrectly: where you have $-3xy^{-2}\frac{dy}{dx}$ inside the brackets, it should be $-3xy^{-2}\left(\frac{dy}{dx}\right)^2$ because $\frac{d}{dx}y^{-1}=-y^{-2}\frac{dy}{dx}$.

Because this term is zero, you go on to obtain the correct value, but I think you have been marked fairly.