Step-by-step parametric differentiation:
Given $x = a\sin^3 t$ and $y = b\cos^3 t$:

• First derivatives with respect to $t$:
  $\frac{dx}{dt} = 3a\sin^2 t \cos t$, and $\frac{dy}{dt} = -3b\cos^2 t \sin t$.
• First derivative $\frac{dy}{dx}$:
  $\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{-3b\cos^2 t \sin t}{3a\sin^2 t \cos t} = -\frac{b}{a} \cot t$.
• Second derivative $\frac{d^2 y}{dx^2}$:
  $\frac{d^2 y}{dx^2} = \frac{d}{dt}\left(-\frac{b}{a}\cot t\right) \cdot \frac{dt}{dx} = \left(\frac{b}{a} \operatorname{cosec}^2 t\right) \cdot \frac{1}{3a\sin^2 t \cos t}$
  $= \frac{b}{3a^2} \frac{\operatorname{cosec}^2 t}{\sin^2 t \cos t} = \frac{b}{3a^2} \operatorname{cosec}^4 t \sec t$.
  
  Option breakdown:
• Inverts the trigonometric functions into numerator powers.
• $\frac{b}{3a^2}\operatorname{cosec}^4 t \sec t$ is the exact second derivative.
• Inverts the constant coefficient to $a/(3b^2)$.
• Swaps the coefficients $a$ and $b$.

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 $t=0$.

Mathematical definition of a cusp versus a vertical tangent:
For a continuous curve $y = f(x)$ at $x = c$:

• Vertical Tangent: Both one-sided limits of the derivative approach the same signed infinity (both $+\infty$ or both $-\infty$). Examples: $f(x) = x^{1/3}$ has $\lim_{x \to 0} f'(x) = +\infty$.
• Cusp: The one-sided limits of the derivative approach opposite signed infinities. That is, as $x \to c^-$, $f'(x) \to +\infty$ and as $x \to c^+$, $f'(x) \to -\infty$ (or vice versa). Example: $f(x) = 1 - x^{2/3}$ or $f(x) = x^{2/3}$.
  
  Option breakdown:
• Both limits tending to $+\infty$ defines an inflection point with a vertical tangent line, not a cusp.
• Both limits tending to $-\infty$ also defines a vertical tangent, not a cusp.
• Opposite infinities ($+\infty$ and $-\infty$) precisely defines a cusp where the curve reverses direction sharply.
• Finite unequal limits define a corner, not a cusp.