Derivative of y=(x²+1)(5-x)

1. Apply the product rule:

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

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

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

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

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

      The result is: $2 x$

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

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

      1. The derivative of the constant $5$ is zero.

      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: $-1$

      The result is: $-1$

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

2. Now simplify:

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

The answer is:

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