Derivative of f(x)=sin(x²+2x)

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:

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

4. Now simplify:

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

The answer is:

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