why implicit partial differentiation and explicit partial differentiation of z with respect to x give different results?

Question

I had a calculus exam and the question was to take the partial of z with respect ot x if: $$xe^y+yz+ze^x=1$$ I wrote z in terms of x and y: $$z= \frac{1-xe^y}{y+e^x}$$ I took the partial of the both sides with respect to x and I found: $$ \frac{\partial z}{\partial x}=\frac{e^y(y+e^x)-e^x(1-xe^y)}{(y+e^x)^2}$$ however, our professor used the implicit partial differention formula which leads to: $$ \frac{\partial z}{\partial x}=-\frac{e^y+ze^x}{(y+e^x)}$$ it is clear that the two are different, should the professor give me credits? Besides, could you please explain me what is wrong with my solution. thanks i advance

Answer 1

Your solution is wrong. You lost a minus sign. It should be$$\frac{\partial z}{\partial x}=\frac{-e^y(y+e^x)-e^x(1-xe^y)}{(y+e^x)^2} .\tag0$$ If you take $$z= \frac{1-xe^y}{y+e^x}\tag1$$ and substitute into the professor's answer $$-\frac{e^y+ze^x}{(y+e^x)} \tag2$$ then it agrees with $(0)$