We are asked to differentiate the function $y = \sqrt[3]{(ax+b)^2}$ with respect to $x$.

1. Problem Description

We are asked to differentiate the function $y = \sqrt[3]{(ax+b)^2}$ with respect to $x$.

2. Solution Steps

First, we can rewrite the function using fractional exponents:

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

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

In our case, $f(u) = u^{2/3}$ and $g(x) = ax+b$.

The derivative of $f(u)$ with respect to $u$ is:

$f'(u) = \frac{2}{3} u^{\frac{2}{3} - 1} = \frac{2}{3} u^{-\frac{1}{3}}$

The derivative of $g(x)$ with respect to $x$ is:

$g'(x) = a$

Applying the chain rule, we have:

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

3. Final Answer

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