Problem
Find the derivative, using implicit differentiation: $$2x^3=(3xy+1)^2$$
Progress
Used the chain rule for the derivative $(3xy+1)^2$. Do I move the $2x^3$ over once I get its derivative, which is $6x^2$?
I know the answer is $(-3y^2x-y+x^2)/(x(3yx+1))$, but I am have trouble solving this problem.
First, it may be helpful to look at patrickJMT's video on implicit differentiation.
Being careful to use the chain rule when needed, take the derivative of both sides with respect to $x$,
$$6x^2 = 2(3xy+1)\left(3y + 3x \frac{dy}{dx}\right).$$
Expand the RHS to obtain $$6x^2 = 18x^2 y\frac{dy}{dx} + 6 x\frac{dy}{dx} + 18xy^2 + 6y.$$ Rearranging, $$6x^2 - 18xy^2 - 6y = \frac{dy}{dx}(18x^2y + 6x).$$ Which gives
$$\frac{dy}{dx} = \frac{6x^2 - 18xy^2 - 6y}{18x^2y + 6x}.$$
Of course, you could clean this up a bit if desired.