Question about implicit differentiation, finding $\frac{dx}{dt}$

Question

Suppose $xy=2$ and $\dfrac{dy}{dt}=2$. Find $\dfrac{dx}{dt}$ when $x=3$.

I don't know how to solve this. I know that if I'm differentiating with respect to time $t$, then the chain rule states that $\dfrac{d}{dt}(xy)=\dfrac{dy}{dt}x+\dfrac{dx}{dt}y$, but I don't know how to proceed from here...

Thank you.

Answer 1

Since $xy=2$ you have that $$0=\frac{d{\bf{2}}}{dt}=x\frac{dy}{dt}+y\frac{dx}{dt}.$$ Now, you know $x=3$ (and so $y=2/3$) and $\frac{dy}{dt}=2.$ Thus, you can get $\frac{dx}{dt}.$

Answer 2

Hint: $$\frac{dx}{dt} = \frac{dx}{dy}\cdot \frac{dy}{dt}$$ You already have $\frac{dy}{dt}$ and can solve for $\frac{dx}{dy}$ by noting that $y = \frac{2}{x}$