Understanding Perfect Differentials

[hhhmortal]

Homework Statement

Hi, I have an exam tomorrow and I'm trying to do the following question which I've almost solved:

Find the perfect differential of the following:

[(x² + 2xy) / (x + y)² ]dx - [x² / (x + y)² ]dy

I differentiated the first term with respect to 'y' and then the second term with respect to x, equated them but they weren't the same, what could I be doing wrong?

The problem is I can't seem to get both terms to equal each other unless i cancel the denominator..but why can't i get them to equal each other if i differentiate with the denominators too? Should be the same...?

Homework Equations

The Attempt at a Solution

 

[quasar987]

I get that they are not equal either.

 

[hhhmortal]

I get that they are not equal either.

If I were to cancel the denominator and then differentiate they actually equal each other. I'm guessing this question was meant to trick me into cancelling it then?

 

[quasar987]

I don't think so. Go see your prof about this.

 

[HallsofIvy]

What does "Find the perfect differential of the following" mean?

I know what determining whether a differential is a perfect differential or not means but this seems to imply that there is some perfect differential associated with this differential. My first guess my be to change it in someway so that this becomes a perfect differential- for example, multiplying the entire expression by $(x+ y)^2$ makes it a perfect differential- but that is certainly not unique so "the" perfect differential would not apply.