Answer
$$y = - \frac{1}{3}x - 8$$
Work Step by Step
$$\eqalign{
& {x^2} = - 6y,{\text{ point }}\left( { - 6, - 6} \right) \cr
& {\text{By the implicit differentiation}} \cr
& \frac{d}{{dx}}\left[ {{x^2}} \right] = \frac{d}{{dx}}\left[ { - 6y} \right] \cr
& 2x = - 6y\frac{{dy}}{{dx}} \cr
& {\text{Solve for }}\frac{{dy}}{{dx}} \cr
& \frac{{dy}}{{dx}} = \frac{{2x}}{{ - 6y}} \cr
& \frac{{dy}}{{dx}} = - \frac{x}{{3y}} \cr
& {\text{Calculate the slope at the point }}\left( { - 6, - 6} \right) \cr
& m = {\left. {\frac{{dy}}{{dx}}} \right|_{\left( { - 6, - 6} \right)}} = - \frac{{ - 6}}{{3\left( { - 6} \right)}} \cr
& m = - \frac{1}{3} \cr
& {\text{Using the point - slope form }}y - {y_1} = m\left( {x - {x_1}} \right),{\text{ with }}\underbrace {\left( {-6, - 6} \right)}_{\left( {{x_1},{y_1}} \right)} \cr
& y - {y_1} = m\left( {x - {x_1}} \right) \cr
& y - \left( { - 6} \right) = - \frac{1}{3}\left( {x + 6} \right) \cr
& y + 6 = - \frac{1}{3}x - 2 \cr
& y = - \frac{1}{3}x - 8 \cr} $$
