Question

find dy.
y = 2 ln (3 + x³)
dy = □ 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:

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