Derivative of cos(x^2+1)

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

2. The derivative of cosine is negative sine:

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

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{2} + 1\right)$:

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

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

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

      The result is: $2 x$

   The result of the chain rule is:

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

4. Now simplify:

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

The answer is: