Derivative of y=sin²4x+1/2cos8x

1. Differentiate $\sin^{24}{\left(x \right)} + \frac{\cos{\left(8 x \right)}}{2}$ term by term:

   1. Let $u = \sin{\left(x \right)}$.

   2. Apply the power rule: $u^{24}$ goes to $24 u^{23}$

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$:

      1. The derivative of sine is cosine:

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

      The result of the chain rule is:

      $$24 \sin^{23}{\left(x \right)} \cos{\left(x \right)}$$

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

      1. Let $u = 8 x$.

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

         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:

         $$- 8 \sin{\left(8 x \right)}$$

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

   The result is: $24 \sin^{23}{\left(x \right)} \cos{\left(x \right)} - 4 \sin{\left(8 x \right)}$

The answer is:

$$24 \sin^{23}{\left(x \right)} \cos{\left(x \right)} - 4 \sin{\left(8 x \right)}$$