Answer
We evaluate the gradient of $\frac{f}{g}$ and prove the Quotient Rule:
$\nabla \left( {\frac{f}{g}} \right) = \frac{{g\nabla f - f\nabla g}}{{{g^2}}}$
Work Step by Step
Let $f$ and $g$ be functions of $x$ and $y$. Note that the methods used here are applicable to the case where $f$ and $g$ are functions of $x$, $y$ and $z$.
We evaluate the gradient of $\frac{f}{g}$, that is:
$\nabla \left( {\frac{f}{g}} \right) = \left( {\frac{\partial }{{\partial x}}\left( {\frac{f}{g}} \right),\frac{{\partial f}}{{\partial y}}\left( {\frac{f}{g}} \right)} \right)$
$ = \left( {\frac{{g\partial f/\partial x - f\partial g/\partial x}}{{{g^2}}},\frac{{g\partial f/\partial y - f\partial g/\partial y}}{{{g^2}}}} \right)$
$ = \frac{1}{{{g^2}}}\left( {g\frac{{\partial f}}{{\partial x}} - f\frac{{\partial g}}{{\partial x}},g\frac{{\partial f}}{{\partial y}} - f\frac{{\partial g}}{{\partial y}}} \right)$
$ = \frac{1}{{{g^2}}}\left( {g\left( {\frac{{\partial f}}{{\partial x}},\frac{{\partial f}}{{\partial y}}} \right) - f\left( {\frac{{\partial g}}{{\partial x}},\frac{{\partial g}}{{\partial y}}} \right)} \right)$
Since the gradients of $f$ and $g$ are
$\nabla f = \left( {\frac{{\partial f}}{{\partial x}},\frac{{\partial f}}{{\partial y}}} \right)$, ${\ \ \ }$ $\nabla g = \left( {\frac{{\partial g}}{{\partial x}},\frac{{\partial g}}{{\partial y}}} \right)$
So, $\nabla \left( {\frac{f}{g}} \right) = \frac{1}{{{g^2}}}\left( {g\nabla f - f\nabla g} \right)$. Hence,
$\nabla \left( {\frac{f}{g}} \right) = \frac{{g\nabla f - f\nabla g}}{{{g^2}}}$
