Exercise $3-$Partial derivatives

Find the domain and the first partial derivatives of the following functions $.$

$(1)f_1(x,y)=\sqrt[\phantom{2}]{x}ln(y),$

$(2)f_2(x,y)=\sqrt[\phantom{2}]{x^2+x+2}ln(y^2+1),$

$(3)f_3(x,y)=ln(x^2+x^3{y}^4),$

$(4)f_4(x,y)=\frac{e^{x+y}}{x+y^2}.$