Derivative of x(x^2-5x+1)

1. Apply the product rule:

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

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

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

   $g{\left(x \right)} = \left(x^{2} - 5 x\right) + 1$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Differentiate $\left(x^{2} - 5 x\right) + 1$ term by term:

      1. Differentiate $x^{2} - 5 x$ 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: $-5$

         The result is: $2 x - 5$

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

      The result is: $2 x - 5$

   The result is: $x^{2} + x \left(2 x - 5\right) - 5 x + 1$

2. Now simplify:

   $$3 x^{2} - 10 x + 1$$

The answer is:

$$3 x^{2} - 10 x + 1$$