Derivative of (sin^2)*(cos3x)

1. Apply the product rule:

   $$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$$

   $f{\left(x \right)} = \sin^{2}{\left(x \right)}$; to find $\frac{d}{d x} f{\left(x \right)}$:

   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:

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

   $g{\left(x \right)} = \cos{\left(3 x \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = 3 x$.

   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 $\frac{d}{d x} 3 x$:

      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: $3$

      The result of the chain rule is:

      $$- 3 \sin{\left(3 x \right)}$$

   The result is: $- 3 \sin^{2}{\left(x \right)} \sin{\left(3 x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(3 x \right)}$

2. Now simplify:

   $$\frac{\left(- \cos{\left(2 x \right)} + 5 \cos{\left(4 x \right)}\right) \sin{\left(x \right)}}{2}$$

The answer is:

$$\frac{\left(- \cos{\left(2 x \right)} + 5 \cos{\left(4 x \right)}\right) \sin{\left(x \right)}}{2}$$