Simplify the following expression. \[ \frac{d}{d x} \int_{2}^{x^{3}} \frac{d p}{p^{2}} \]

To simplify the expression \( \frac{d}{d x} \int_{2}^{x^{3}} \frac{d p}{p^{2}} \), we can apply the Fundamental Theorem of Calculus along with the Chain Rule. First, we consider the integral: \[ \int_{2}^{x^{3}} \frac{d p}{p^{2}}. \] The integrand \( \frac{1}{p^{2}} \) is straightforward to integrate. The indefinite integral is: \[ -\frac{1}{p} + C. \] Evaluating the definite integral gives: \[ \int_{2}^{x^{3}} \frac{d p}{p^{2}} = -\frac{1}{x^{3}} + \frac{1}{2}. \] Now we simplify our original expression: \[ \frac{d}{d x} \left(-\frac{1}{x^{3}} + \frac{1}{2}\right). \] Next, we differentiate this: \[ \frac{d}{d x} \left(-\frac{1}{x^{3}} + \frac{1}{2}\right) = \frac{d}{d x} \left(-\frac{1}{x^{3}}\right) + \frac{d}{d x} \left(\frac{1}{2}\right). \] The constant \( \frac{1}{2} \) disappears upon differentiation, and we apply the power rule for the first term: \[ \frac{d}{d x} \left(-\frac{1}{x^{3}}\right) = 3x^{-4} = \frac{3}{x^{4}}. \] Thus, the simplified expression is: \[ \frac{3}{x^{4}}. \]