Derivative of y=sin4x-2x+3

1. Differentiate $\left(- 2 x + \sin{\left(4 x \right)}\right) + 3$ term by term:

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

      1. Let $u = 4 x$.

      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 $\frac{d}{d x} 4 x$:

         1. 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 of the chain rule is:

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

      4. 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: $-2$

      The result is: $4 \cos{\left(4 x \right)} - 2$

   2. The derivative of the constant $3$ is zero.

   The result is: $4 \cos{\left(4 x \right)} - 2$

The answer is:

$$4 \cos{\left(4 x \right)} - 2$$