Derivative of 3sinx*cos3x+sin3x*cosx

1. Differentiate $3 \sin{\left(x \right)} \cos{\left(3 x \right)} + \sin{\left(3 x \right)} \cos{\left(x \right)}$ term by term:

   1. Apply the product rule:

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

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

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

         1. The derivative of sine is cosine:

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

         So, the result is: $3 \cos{\left(x \right)}$

      $g{\left(x \right)} = \cos{\left(3 x \right)}$; 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: $- 9 \sin{\left(x \right)} \sin{\left(3 x \right)} + 3 \cos{\left(x \right)} \cos{\left(3 x \right)}$

   2. Apply the product rule:

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

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

      1. Let $u = 3 x$.

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

         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:

         $$3 \cos{\left(3 x \right)}$$

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

      1. The derivative of cosine is negative sine:

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

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

   The result is: $- 10 \sin{\left(x \right)} \sin{\left(3 x \right)} + 6 \cos{\left(x \right)} \cos{\left(3 x \right)}$

2. Now simplify:

   $$- 2 \cos{\left(2 x \right)} + 8 \cos{\left(4 x \right)}$$

The answer is:

$$- 2 \cos{\left(2 x \right)} + 8 \cos{\left(4 x \right)}$$