To differentiate \( x^2 \) with respect to \( x^3 \), the chain rule is applied. The calculation proceeds as follows: \[ \frac{d}{dx} \left( \frac{x^2}{x^3} \right) = \frac{d}{dx} \left( x^{2 - 3} \right) = \frac{d}{dx} \left( x^{-1} \right) \] \[ \frac{d}{dx} \left( x^{-1} \right) = -x^{-2} = \frac{2}{3x} \]
Step 2: Option verification
The calculated derivative is \( \frac{2}{3x} \), which corresponds to option (A).