Derivative of cos(1-4x)

1. Let $u = 1 - 4 x$.

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 $\frac{d}{d x} \left(1 - 4 x\right)$:

   1. Differentiate $1 - 4 x$ 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: $-4$

      The result is: $-4$

   The result of the chain rule is:

   $$- 4 \sin{\left(4 x - 1 \right)}$$

The answer is:

$$- 4 \sin{\left(4 x - 1 \right)}$$