Derivative of (x-1)*(x-2)*(x-3)*(x-4)*(x-5)

1. Apply the product rule:

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

   $f{\left(x \right)} = \left(x - 2\right) \left(x - 1\right) \left(x - 3\right) \left(x - 4\right)$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Apply the product rule:

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

      $f{\left(x \right)} = \left(x - 2\right) \left(x - 1\right) \left(x - 3\right)$; to find $\frac{d}{d x} f{\left(x \right)}$:

      1. Apply the product rule:

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

         $f{\left(x \right)} = \left(x - 2\right) \left(x - 1\right)$; to find $\frac{d}{d x} f{\left(x \right)}$:

         1. Apply the product rule:

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

            $f{\left(x \right)} = x - 1$; to find $\frac{d}{d x} f{\left(x \right)}$:

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

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

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

               The result is: $1$

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

            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 is: $2 x - 3$

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

         1. Differentiate $x - 3$ term by term:

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

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

            The result is: $1$

         The result is: $\left(x - 3\right) \left(2 x - 3\right) + \left(x - 2\right) \left(x - 1\right)$

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

      1. Differentiate $x - 4$ term by term:

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

         2. The derivative of the constant $-4$ is zero.

         The result is: $1$

      The result is: $\left(x - 2\right) \left(x - 1\right) \left(x - 3\right) + \left(x - 4\right) \left(\left(x - 3\right) \left(2 x - 3\right) + \left(x - 2\right) \left(x - 1\right)\right)$

   $g{\left(x \right)} = x - 5$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Differentiate $x - 5$ term by term:

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

      2. The derivative of the constant $-5$ is zero.

      The result is: $1$

   The result is: $\left(x - 2\right) \left(x - 1\right) \left(x - 3\right) \left(x - 4\right) + \left(x - 5\right) \left(\left(x - 2\right) \left(x - 1\right) \left(x - 3\right) + \left(x - 4\right) \left(\left(x - 3\right) \left(2 x - 3\right) + \left(x - 2\right) \left(x - 1\right)\right)\right)$

2. Now simplify:

   $$5 x^{4} - 60 x^{3} + 255 x^{2} - 450 x + 274$$

The answer is:

$$5 x^{4} - 60 x^{3} + 255 x^{2} - 450 x + 274$$