Implicitly differentiating term by term, we find
∂ x ∂ x ( y z ) + ∂ z ∂ x ( x y ) + ∂ x ∂ x + 3 z 2 ∂ z ∂ x = 0 ∂ z ∂ x ( x y + 3 z 2 ) = − ( y z + 1 ) ∂ z ∂ x = − y z + 1 x y + 3 z 2 {\displaystyle {\begin{aligned}{\frac {\partial x}{\partial x}}(yz)+{\frac {\partial z}{\partial x}}(xy)+{\frac {\partial x}{\partial x}}+3z^{2}{\frac {\partial z}{\partial x}}&=0\\{\frac {\partial z}{\partial x}}(xy+3z^{2})&=-(yz+1)\\{\frac {\partial z}{\partial x}}&=-{\frac {yz+1}{xy+3z^{2}}}\end{aligned}}}
