Derivative of 8*cos(x)+sin(7*x)-16*x

1. Differentiate $- 16 x + \left(\sin{\left(7 x \right)} + 8 \cos{\left(x \right)}\right)$ term by term:

   1. Differentiate $\sin{\left(7 x \right)} + 8 \cos{\left(x \right)}$ term by term:

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

         1. The derivative of cosine is negative sine:

            $$\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$$

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

      2. Let $u = 7 x$.

      3. The derivative of sine is cosine:

         $$\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$$

      4. Then, apply the chain rule. Multiply by $\frac{d}{d x} 7 x$:

         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: $7$

         The result of the chain rule is:

         $$7 \cos{\left(7 x \right)}$$

      The result is: $- 8 \sin{\left(x \right)} + 7 \cos{\left(7 x \right)}$

   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: $-16$

   The result is: $- 8 \sin{\left(x \right)} + 7 \cos{\left(7 x \right)} - 16$

The answer is:

$$- 8 \sin{\left(x \right)} + 7 \cos{\left(7 x \right)} - 16$$