If the relation $F(x,y,z)=z^4+x+ysinx+xlnz=0$ defines $z$ as a differentiable function of $x$ and $y.$ Using the implicit differentiation at the point $(0,0,1),$ we have:

None of them

del$\frac{z}{d}$elx$=-\frac{1}{4}$ and del$\frac{z}{d}$ely$=0$

del$\frac{z}{d}$elx$=\frac{1}{4}$ and del$\frac{z}{d}$ely$=0$

del$\frac{z}{d}$elx$=0$ and del$\frac{z}{d}$ely$=\frac{1}{4}$