The problem asks us to differentiate the function $y = \sqrt[3]{(ax+b)^2}$ with respect to $x$.

1. Problem Description

The problem asks us to differentiate the function $y = \sqrt[3]{(ax+b)^2}$ with respect to $x$.

2. Solution Steps

First, rewrite the function using fractional exponents:

$y = (ax+b)^{\frac{2}{3}}$

Now, we differentiate $y$ with respect to $x$ using the chain rule. The chain rule states that if $y = f(u)$ and $u = g(x)$, then $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$.

Let $u = ax + b$. Then $y = u^{\frac{2}{3}}$.

$\frac{dy}{du} = \frac{2}{3}u^{\frac{2}{3}-1} = \frac{2}{3}u^{-\frac{1}{3}}$

$\frac{du}{dx} = \frac{d}{dx}(ax+b) = a$

Using the chain rule:

$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = \frac{2}{3}u^{-\frac{1}{3}} \cdot a = \frac{2a}{3}(ax+b)^{-\frac{1}{3}}$

Rewrite the expression with the cube root:

$\frac{dy}{dx} = \frac{2a}{3\sqrt[3]{ax+b}}$

3. Final Answer

$\frac{dy}{dx} = \frac{2a}{3\sqrt[3]{ax+b}}$