Multivariable chain rule evaluation:
$\frac{d\omega}{dt} = \frac{\partial\omega}{\partial x}\frac{dx}{dt} + \frac{\partial\omega}{\partial y}\frac{dy}{dt}$.
Here $\omega = x^2 y - y^2$:
$\frac{\partial\omega}{\partial x} = 2xy$, $\frac{\partial\omega}{\partial y} = x^2 - 2y$.
Given $x = \sin t \implies \frac{dx}{dt} = \cos t$.
Given $y = e^t \implies \frac{dy}{dt} = e^t$.
At $t = 0$:
$x = \sin 0 = 0$, $y = e^0 = 1$.
$\frac{dx}{dt} = \cos 0 = 1$, $\frac{dy}{dt} = e^0 = 1$.
Substitute these values:
$\frac{\partial\omega}{\partial x} = 2(0)(1) = 0$.
$\frac{\partial\omega}{\partial y} = 0^2 - 2(1) = -2$.
$\frac{d\omega}{dt} = (0)(1) + (-2)(1) = -2$.

Option breakdown:

• $0$ forgets the contribution from the $\partial\omega/\partial y$ term.
• $-1$ is an incorrect calculation.
• $2$ has an incorrect positive sign.
• $-2$ is the correct total derivative evaluated at $t=0$.