Derivative of 2sin(3-4*x)

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

   1. Let $u = 3 - 4 x$.

   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 $\frac{d}{d x} \left(3 - 4 x\right)$:

      1. Differentiate $3 - 4 x$ term by term:

         1. The derivative of the constant $3$ 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:

      $$- 4 \cos{\left(4 x - 3 \right)}$$

   So, the result is: $- 8 \cos{\left(4 x - 3 \right)}$

The answer is:

$$- 8 \cos{\left(4 x - 3 \right)}$$