Find gradient of this implicit function How to find a gradient of this implicit function?
$$
xz+yz^2-3xy-3=0
$$
EDIT: Abhinav pointed out a mistake, which has been corrected for posterity.

To find $\frac{\partial z}{\partial x}$ by implicit differentiation means to differentiate both sides of the equation with respect to $x$, while remembering that $z$ is implicitly a function of $x$ (we didn't write it with function notation such as $z(x)$).  Don't forget to use the product rule when differentiating a product!
$$
\begin{align}
xz + yz^2 - 3xy - 3 &amp;= 0 \\
\tfrac{\partial}{\partial x} \left[ xz + yz^2 - 3xy - 3 \right] &amp;= \tfrac{\partial}{\partial x} \left[ 0 \right] \\
\tfrac{\partial}{\partial x} \left[ xz \right] + \tfrac{\partial}{\partial x} \left[ yz^2 \right] - \tfrac{\partial}{\partial x} \left[ 3xy \right] - \tfrac{\partial}{\partial x} \left[ 3 \right] &amp;= 0 \\
\left[ 1 \cdot z + x \cdot \tfrac{\partial z}{\partial x} \right] + y \cdot 2z \tfrac{\partial z}{\partial x} - 3y - 0 &amp;= 0 \\
z + x \tfrac{\partial z}{\partial x} + 2yz \tfrac{\partial z}{\partial x} - 3y &amp;= 0
\end{align}
$$
This yields
$$
\frac{\partial z}{\partial x} = \frac{3y - z}{x + 2yz}.
$$

Try to find $\tfrac{\partial z}{\partial y}$ in an analogous way, and post your result in the comments.
By using implicit differentation, determine the gradient of curve $3xy+y^2= -2$ at the point $(1,-3)$