So I've been trying to implicitly differentiate $x^2 y^3 = 3$ which if I use product rule I get $$ -\frac{2y}{3x} $$ but when I move $x^2$ to the right hand side and differentiate I get $$ -\frac{2}{x^3 y^2} $$ which is completely different answer and I do not get what I did wrong.
Help Please.
Beginning with the given equation $$ x^2 y^3 = 3, $$ we can modify the implicit derivative expression to look like the other:
$$ -\frac{2y}{3x} = -\frac{2y}{3x} \color{blue}{{}\cdot \frac{x^2 y^2}{x^2 y^2}} = -\frac{2x^2 y^3}{3x^3 y^2} = -\frac{2 \cdot3}{3x^3 y^2} = -\frac{2}{x^3 y^2}. $$