Partial Derivative Questions

Hi i've attempted a few problems from my textbook which i've understood but i've come across 2 questions which have completly got my confused, and wouldnt mind some assistance :p.

1) Prove that if $\displaystyle z=2xy+x.f\frac{y}{x}$ then $\displaystyle x\frac{{\delta}z}{{\delta}x}+y\frac{{\delta}z}{{\d elta}y}=z+2xy$

2)If f=F(x,y) and $\displaystyle x=re^\theta$, $\displaystyle y=re^{-\theta}$ prove that $\displaystyle 2x\frac{{\delta}f}{{\delta}x}=r\frac{{\delta}f}{{\ delta}r}+{{\delta}f}{{\delta}\theta}$ and that $\displaystyle 2y\frac{{\delta}f}{{\delta}y}=r\frac{{\delta}f}{{\ delta}r}-{{\delta}f}{{\delta}\theta}$