I'm taking an elementary differential equations class and we got our first exam back and...

I'm taking an elementary differential equations class and we got our first exam back and I don't understand why I was wrong on one of the questions. I got a 96, and the professor said there were quite a few d's and f's, so I didn't want to quibble about this.
Anyway, the problem was ${\displaystyle \frac{dy}{dx}}={\displaystyle \frac{1}{x+y+2}}$ and I chose the sub $u=x+y+2$, with ${\displaystyle \frac{dy}{dx}}={\displaystyle \frac{dy}{du}}{\displaystyle \frac{du}{dx}}={\displaystyle \frac{dy}{du}}(1+{\displaystyle \frac{1}{u}})$ and then plugged in the expression for ${\displaystyle \frac{dy}{dx}}$, separated the equation and solved for y in terms of u and then substituted back to get $y=\ln (x+y+3)+c.$ The professor took issue with my expression for ${\displaystyle \frac{dy}{dx}}$, saying that I couldn't differentiate y w.r.t. u if y is part of u. Is this correct?