Derivative of 2*sin^5(x)

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: $u^{5}$ goes to $5 u^{4}$

   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:

      $$5 \sin^{4}{\left(x \right)} \cos{\left(x \right)}$$

   So, the result is: $10 \sin^{4}{\left(x \right)} \cos{\left(x \right)}$

The answer is:

$$10 \sin^{4}{\left(x \right)} \cos{\left(x \right)}$$