Question: User: 𝑦=𝑥(2𝑥+1)2 Solve the derivative for each of the following functions:
1. **State the problem:** We need to find the derivative of the function $$y = x(2x+1)^2$$.
2. **Recall the formula:** To differentiate a product of two functions, use the product rule: $$\frac{d}{dx}[u \cdot v] = u'v + uv'$$.
3. **Identify parts:** Let $$u = x$$ and $$v = (2x+1)^2$$.
4. **Differentiate each part:**
- $$u' = \frac{d}{dx}[x] = 1$$
- To find $$v'$$, use the chain rule: $$v = (2x+1)^2$$, so
$$v' = 2(2x+1) \cdot \frac{d}{dx}[2x+1] = 2(2x+1) \cdot 2 = 4(2x+1)$$.
5. **Apply product rule:**
$$\frac{dy}{dx} = u'v + uv' = 1 \cdot (2x+1)^2 + x \cdot 4(2x+1)$$.
6. **Simplify:**
$$\frac{dy}{dx} = (2x+1)^2 + 4x(2x+1)$$.
7. **Factor common terms:**
$$\frac{dy}{dx} = (2x+1)[(2x+1) + 4x] = (2x+1)(2x+1 + 4x) = (2x+1)(6x+1)$$.
**Final answer:**
$$\boxed{\frac{dy}{dx} = (2x+1)(6x+1)}$$