An object is moving in the clockwise direction around the unit circle $x^2+y^2=1$. As it passes through the point $(\frac{1}{2},\frac{\sqrt{2}}{2})$, it’s y-coordinate is decreasing at the rate of 3 units per second. The rate at which the x-coordinate changes at this point is (in units per second)

$3\sqrt{3}$

We find $\frac{dx}{dt}$, when $x=\frac{1}{2}$ and $y=\frac{\sqrt{3}}{2}$ given that $\frac{dy}{dt}= −3$ units/s and $x^2+y^2=1$

Differentiating $x^2+y^2=1$, we have $2x\frac{dx}{dt}+ 2y\frac{dy}{dt} = 0$

Putting $x=\frac{1}{2}$, $y=\frac{\sqrt{3}}{2}$ and $\frac{dy}{dt}=-3$, we have, $\frac{1}{2}\frac{dx}{dt}+\frac{\sqrt{3}}{2}(-3)=0⇒\frac{dx}{dt}=3\sqrt{3}$