Derivative of 3sin^2x-sin^3x

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

   1. 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: $u^{2}$ 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:

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

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

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

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

2. Now simplify:

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

The answer is: