Derivative of sin3x+2sin^3*(x)

1. Differentiate $2 \sin^{3}{\left(x \right)} + \sin{\left(3 x \right)}$ term by term:

   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:

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

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

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

      2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$

      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:

         $$3 \sin^{2}{\left(x \right)} \cos{\left(x \right)}$$

      So, the result is: $6 \sin^{2}{\left(x \right)} \cos{\left(x \right)}$

   The result is: $6 \sin^{2}{\left(x \right)} \cos{\left(x \right)} + 3 \cos{\left(3 x \right)}$

2. Now simplify:

   $$\frac{3 \cos{\left(x \right)}}{2} + \frac{3 \cos{\left(3 x \right)}}{2}$$

The answer is:

$$\frac{3 \cos{\left(x \right)}}{2} + \frac{3 \cos{\left(3 x \right)}}{2}$$