Names:

Consider the plot of "all pairs", (x ,y) which satisfy y^3 - xy = -6:

There are 3 such pairs with x = 7. By using the Solve features of your calculator you can find them. Here are the three y values (compliments of Maple):

> solve(y^3-7*y+6=0,{y});

Theoretically, we could take the equation y^3 - x*y = -6 and solve it for y. This would yield 3 formulas expressing y in terms of x, i.e. we can view y as a function of x (or many functions of x), as in y = f(x). In this case, we can actually do this (with the help of Maple), but the answers are extremely complicated (and so omitted from the HTML file).

> map(simplify,{solve(y^3-x*y+6=0,y)});
 

From a graphical perspective however, the 3 functions are easy to see:

Near x = 7, which of f '(x), g'(x) and h'(x) are positive or negative? Which is closest to zero?

Using the zoomed in view below, estimate f '(7) and g'(7).

Picking various values of x near 7, and solving for y values near -3 (on the graph of h(x) ), gives:

Use the table above to estimate dy/dx at the point (7, -3).

Now, treat y as a function of x, calculate dy/dx using implicit differentiation, and substitute in x = 7, with y = 1, 2 and -3. Are these values close to your estimates?

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