Derivative of y=sin³(x²+2x)

1. Let $u = \sin{\left(x^{2} + 2 x \right)}$.

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

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

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

   2. The derivative of sine is cosine:

      $$\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$$

   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:

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

   The result of the chain rule is:

   $$3 \cdot \left(2 x + 2\right) \sin^{2}{\left(x^{2} + 2 x \right)} \cos{\left(x^{2} + 2 x \right)}$$

4. Now simplify:

   $$\left(6 x + 6\right) \sin^{2}{\left(x \left(x + 2\right) \right)} \cos{\left(x \left(x + 2\right) \right)}$$

The answer is: