Derivative of sin(12(x^2)+x)

1. Let $u = 12 x^{2} + x$.

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 $\frac{d}{d x} \left(12 x^{2} + x\right)$:

   1. Differentiate $12 x^{2} + x$ term by term:

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

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

         So, the result is: $24 x$

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

      The result is: $24 x + 1$

   The result of the chain rule is:

   $$\left(24 x + 1\right) \cos{\left(12 x^{2} + x \right)}$$

4. Now simplify:

   $$\left(24 x + 1\right) \cos{\left(x \left(12 x + 1\right) \right)}$$

The answer is: