19. \( \left(-18 m^{2} n\right)^{2}\left(-\frac{1}{6} m n^{2}\right)^{\prime} \)

To simplify the expression \( \left(-18 m^{2} n\right)^{2}\left(-\frac{1}{6} m n^{2}\right)^{\prime} \), we'll tackle it step by step. First, let's simplify \( \left(-18 m^{2} n\right)^{2} \): \[ (-18)^{2} (m^{2})^{2} (n)^{2} = 324 m^{4} n^{2} \] Next, we need to find the derivative of \( -\frac{1}{6} m n^{2} \): Using the product rule, let \( u = -\frac{1}{6} m \) and \( v = n^{2} \): \[ \frac{du}{dx} = -\frac{1}{6}, \quad \frac{dv}{dx} = 2n \frac{dn}{dx} \] Thus, \[ \left(-\frac{1}{6} m n^{2}\right)^{\prime} = u'v + uv' = -\frac{1}{6} n^{2} + (-\frac{1}{6} m)(2n \frac{dn}{dx}) = -\frac{1}{6} n^{2} - \frac{1}{3} mn \frac{dn}{dx} \] Now, combining everything together: \[ 324 m^{4} n^{2} \left(-\frac{1}{6} n^{2} - \frac{1}{3} m n \frac{dn}{dx}\right) \] We can distribute: \[ = 324 m^{4} n^{2} \cdot -\frac{1}{6} n^{2} + 324 m^{4} n^{2} \cdot -\frac{1}{3} m n \frac{dn}{dx} \] \[ = -54 m^{4} n^{4} - 108 m^{5} n^{3} \frac{dn}{dx} \] Thus, the simplified expression is: \[ -54 m^{4} n^{4} - 108 m^{5} n^{3} \frac{dn}{dx} \]