Derivative of sin8x+5cos2x

1. Differentiate $\sin{\left(8 x \right)} + 5 \cos{\left(2 x \right)}$ term by term:

   1. Let $u = 8 x$.

   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 $\frac{d}{d x} 8 x$:

      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: $8$

      The result of the chain rule is:

      $$8 \cos{\left(8 x \right)}$$

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

      1. Let $u = 2 x$.

      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 $\frac{d}{d x} 2 x$:

         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: $2$

         The result of the chain rule is:

         $$- 2 \sin{\left(2 x \right)}$$

      So, the result is: $- 10 \sin{\left(2 x \right)}$

   The result is: $- 10 \sin{\left(2 x \right)} + 8 \cos{\left(8 x \right)}$

The answer is:

$$- 10 \sin{\left(2 x \right)} + 8 \cos{\left(8 x \right)}$$