Derivative of (x^2-2x+3)^5

1. Let $u = \left(x^{2} - 2 x\right) + 3$.

2. Apply the power rule: $u^{5}$ goes to $5 u^{4}$

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

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

      1. Differentiate $x^{2} - 2 x$ term by term:

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

         2. 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 is: $2 x - 2$

      2. The derivative of the constant $3$ is zero.

      The result is: $2 x - 2$

   The result of the chain rule is:

   $$5 \left(2 x - 2\right) \left(\left(x^{2} - 2 x\right) + 3\right)^{4}$$

4. Now simplify:

   $$10 \left(x - 1\right) \left(x^{2} - 2 x + 3\right)^{4}$$

The answer is:

$$10 \left(x - 1\right) \left(x^{2} - 2 x + 3\right)^{4}$$