Answer
a) $\frac{dy}{dt} = 6$
b) $\frac{dx}{dt} = -\frac{1}{3}$
Work Step by Step
a) By the chain rule, $\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}$.
We solve for $\frac{dy}{dx}$ by differentiating the following equation by $\frac{d}{dx}$
$y=3x+5$
$\frac{dy}{dx} = 3$
Thus $\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt} = 3 \cdot 2 = 6$
b)
We now differentiate both sides with respect to $t$:
$y = 3x+5$
$\frac{dy}{dt} = 3\frac{dx}{dt}$
$\frac{dx}{dt} = \frac{dy}{dt}/3=-\frac{1}{3}$
