For example, say you have y^2+5x=6x^3+4
Okay, so we do some implicit derivation and get 18x^2-5 / 2y
Now... how is one to graph that? It is no longer y= or one function equaling another, it is now dy/dx= So... how is one to graph this? I'd like to do so to verify my answers are correct. My professor had no explanation other than "it won't work."
The equation $y^2+5x=6x^3+4$ defines implicitly two functions, namely, $f_1(x)=\sqrt{6x^3-5x+4}$ and $f_2(x)=-\sqrt{6x^3-5x+4}$. From implicit differentiation you know $$\frac{\mathrm d y}{\mathrm dx}=\frac{18x^2-5}{2y}\tag{1}$$ Then you get $$f_1'(x)=\frac{18x^2-5}{2\sqrt{6x^3-5x+4}}\quad\text{and}\quad f_2'(x)=-\frac{18x^2-5}{2\sqrt{6x^3-5x+4}}$$ Both functions can be graphed in the $xy$ plane.