find dy.\ny = 2 ln (3 + x³)\ndy = □ dx

Explanation:

Step1: Differentiate the outer function

The derivative of (y = 2\ln(u)) with respect to (u) (where (u = 3 + x^{3})) is (\frac{dy}{du}=\frac{2}{u}).

Step2: Differentiate the inner function

The derivative of (u = 3 + x^{3}) with respect to (x) is (\frac{du}{dx}=3x^{2}).

Step3: Use the chain rule

By the chain rule (\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}). Substitute (\frac{dy}{du}=\frac{2}{u}) and (\frac{du}{dx}=3x^{2}), and since (u = 3 + x^{3}), we get (\frac{dy}{dx}=\frac{2}{3 + x^{3}}\cdot3x^{2}). Simplify (\frac{2}{3 + x^{3}}\cdot3x^{2}=\frac{6x^{2}}{3 + x^{3}}).

Answer:

(\frac{6x^{2}}{3 + x^{3}})