Derivative of cos(1-7x+4x^2)

1. Let $u = 4 x^{2} + \left(1 - 7 x\right)$.

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 $\frac{d}{d x} \left(4 x^{2} + \left(1 - 7 x\right)\right)$:

   1. Differentiate $4 x^{2} + \left(1 - 7 x\right)$ term by term:

      1. Differentiate $1 - 7 x$ term by term:

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

         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: $-7$

         The result is: $-7$

      2. 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: $8 x$

      The result is: $8 x - 7$

   The result of the chain rule is:

   $$- \left(8 x - 7\right) \sin{\left(4 x^{2} - 7 x + 1 \right)}$$

4. Now simplify:

   $$\left(7 - 8 x\right) \sin{\left(4 x^{2} - 7 x + 1 \right)}$$

The answer is:

$$\left(7 - 8 x\right) \sin{\left(4 x^{2} - 7 x + 1 \right)}$$