Derivative of y=sin(x^2+x+1)

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

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 $\frac{d}{d x} \left(\left(x^{2} + x\right) + 1\right)$:

   1. Differentiate $\left(x^{2} + x\right) + 1$ term by term:

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

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

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

         The result is: $2 x + 1$

      2. The derivative of the constant $1$ is zero.

      The result is: $2 x + 1$

   The result of the chain rule is:

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

4. Now simplify:

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

The answer is: