Answer
$-6$
Work Step by Step
Take the derivative of the given equations and isolate the variables.
$ \dfrac{dx}{dt}=\dfrac{2t+1}{1+3x^{1/2}}$
and $\dfrac{y}{2\sqrt {t+1}}+(\sqrt{t+1}+(t/\sqrt y))\dfrac{dy}{dt}+2 \sqrt y=0$
At $t=0$, we have $x=0$
Then $ \dfrac{dx}{dt}=1$
At $t=0$, we have $y=4$
Then $ \dfrac{dx}{dt}=-6$
Now, Slope: $\dfrac{dy}{dx}=\dfrac{-6}{1}=-6$
