Derivative of 8(sin^2x-cos^2x)

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

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

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

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

      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:

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

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

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

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

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

            1. The derivative of cosine is negative sine:

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

            The result of the chain rule is:

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

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

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

   So, the result is: $32 \sin{\left(x \right)} \cos{\left(x \right)}$

2. Now simplify:

   $$16 \sin{\left(2 x \right)}$$

The answer is:

$$16 \sin{\left(2 x \right)}$$