Derivative of 1+8(x-2)+6(x-2)^2+(x-2)^3

1. Differentiate $\left(x - 2\right)^{3} + \left(6 \left(x - 2\right)^{2} + \left(8 \left(x - 2\right) + 1\right)\right)$ term by term:

   1. Differentiate $6 \left(x - 2\right)^{2} + \left(8 \left(x - 2\right) + 1\right)$ term by term:

      1. Differentiate $8 \left(x - 2\right) + 1$ term by term:

         1. The derivative of the constant $1$ is zero.

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

            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: $1$

            So, the result is: $8$

         The result is: $8$

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

         1. Let $u = x - 2$.

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

         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:

            $$2 x - 4$$

         So, the result is: $12 x - 24$

      The result is: $12 x - 16$

   2. Let $u = x - 2$.

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

   4. 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:

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

   The result is: $12 x + 3 \left(x - 2\right)^{2} - 16$

2. Now simplify:

   $$3 x^{2} - 4$$

The answer is:

$$3 x^{2} - 4$$