
Curve
Given the curve $\displaystyle x^2xy+y^2=9$
(A) write a general expression for the slope of the curve.
(B) Find the coordinates of the points on the curve where the tangents are vertical.
(C) At the point (0,3) find the rate of change in the slope of the curve with respect to x.

Can you do implicit differentiation?.
$\displaystyle 2xxy'y+2yy'=0$
Solve for y'.
For the vertical tangents:
Eliminate all terms without y' and we're left with:
$\displaystyle xy'+2yy'=0$
$\displaystyle y'(x+2y)=0$
$\displaystyle x+2y=0$
$\displaystyle y=\frac{x}{2}$
Sub this back into your original and solve for x to find the points where there are vertical tangents.
The reason this works has to do with the limit as $\displaystyle y'\to {\infty}$

for c the slope = .5 but what does respect to x mean