Derivative of (x-2)^3*sin(2x)

1. Apply the product rule:

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

   $f{\left(x \right)} = \left(x - 2\right)^{3}$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Let $u = x - 2$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x - 2\right)$:

      1. Differentiate $x - 2$ term by term:

         1. Apply the power rule: $x$ goes to $1$

         2. The derivative of the constant $\left(-1\right) 2$ is zero.

         The result is: $1$

      The result of the chain rule is:

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

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

   1. Let $u = 2 x$.

   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 $\frac{d}{d x} 2 x$:

      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: $2$

      The result of the chain rule is:

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

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

2. Now simplify:

   $$\left(x - 2\right)^{2} \left(\left(2 x - 4\right) \cos{\left(2 x \right)} + 3 \sin{\left(2 x \right)}\right)$$

The answer is:

$$\left(x - 2\right)^{2} \left(\left(2 x - 4\right) \cos{\left(2 x \right)} + 3 \sin{\left(2 x \right)}\right)$$