Derivative of (x-2)^4sin6x

1. Apply the product rule:

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

   $f{\left(x \right)} = \left(x - 2\right)^{4}$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Let $u = x - 2$.

   2. Apply the power rule: $u^{4}$ goes to $4 u^{3}$

   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 $-2$ is zero.

         The result is: $1$

      The result of the chain rule is:

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

   $g{\left(x \right)} = \sin{\left(6 x \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = 6 x$.

   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 $\frac{d}{d x} 6 x$:

      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: $6$

      The result of the chain rule is:

      $$6 \cos{\left(6 x \right)}$$

   The result is: $6 \left(x - 2\right)^{4} \cos{\left(6 x \right)} + 4 \left(x - 2\right)^{3} \sin{\left(6 x \right)}$

2. Now simplify:

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

The answer is:

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