I need to find $\operatorname{Im} f$ Let $f$ be analytic such that $\operatorname{Re} ( f'(z))=2y$ and $f(1+i)=2$.  I need to find $\operatorname{Im} f$.
I took $f(z)=u(x,y)+i v(x,y)$ but I don't know $\frac{df(z)}{dz}=$ ? in terms of $u(x,y)$ and $v(x,y)$.
Could any one help me in that step?

Hint: First find $f'(z)$ by the usual method of computing harmonic conjugates. (Just use the Cauchy-Riemann equations.) Then integrate to find $f(z)$ up to a constant of integration. Use your initial condition $f(1+i)=2$ to determine $f(z)$ uniquely. 
