Derivative of y=(x²+2x)⁴

1. Let $u = x^{2} + 2 x$.

2. Apply the power rule: $u^{4}$ goes to $4 u^{3}$

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{2} + 2 x\right)$:

   1. Differentiate $x^{2} + 2 x$ term by term:

      1. Apply the power rule: $x^{2}$ goes to $2 x$

      2. The derivative of a constant times a function is the constant times the derivative of the function.

         1. Apply the power rule: $x$ goes to $1$

         So, the result is: $2$

      The result is: $2 x + 2$

   The result of the chain rule is:

   $$4 \left(2 x + 2\right) \left(x^{2} + 2 x\right)^{3}$$

4. Now simplify:

   $$8 x^{3} \left(x + 1\right) \left(x + 2\right)^{3}$$

The answer is:

$$8 x^{3} \left(x + 1\right) \left(x + 2\right)^{3}$$