Derivative of 5*cos(4*x)-2+5x

1. Differentiate $5 x + \left(5 \cos{\left(4 x \right)} - 2\right)$ term by term:

   1. Differentiate $5 \cos{\left(4 x \right)} - 2$ term by term:

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

         1. Let $u = 4 x$.

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

            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:

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

         So, the result is: $- 20 \sin{\left(4 x \right)}$

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

      The result is: $- 20 \sin{\left(4 x \right)}$

   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: $5$

   The result is: $5 - 20 \sin{\left(4 x \right)}$

The answer is:

$$5 - 20 \sin{\left(4 x \right)}$$