I assume that fy means fᵧ or ∂f/∂y (partial differential or derivative).
If so then we treat x as a constant and differentiate with respect to y only, using the chain rule and product rule.
f(x,y)=y³sin(y²x)ln(4y+x).
Let p=y³, q=sin(y²x), r=ln(4y+x);
∂p/∂y=3y², ∂q/∂y=(cos(y²x))(2xy)=2xycos(y²x), ∂r/∂y=4/(4y+x).
∂f/∂y=∂(pqr)/∂y=pq∂r∂y+r∂(pq)/∂y=pq∂r∂y+r(p∂q/∂y+q∂p/∂y)=
y³sin(y²x)(4/(4y+x))+ln(4y+x)(y³(2xycos(y²x))+sin(y²x)(3y²))=
4y³sin(y²x)/(4y+x)+2xy⁴cos(y²x)ln(4y+x)+3y²sin(y²x)ln(4y+x).