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1. \frac{d}{dx}\left[\frac{x \left(1 - x\right)}{2}\right]

   Differentiate 2 times, one derivative at a time.

2. \frac{\frac{d}{d x} x \left(1 - x\right)}{2}

   Constant multiple rule: pull the constant out.

3. \frac{x \frac{d}{d x} \left(1 - x\right)}{2} + \frac{\left(1 - x\right) \frac{d}{d x} x}{2}

   Product rule with u = x and v = 1 - x: (uv)′ = u′v + uv′.

4. \frac{x \left(\frac{d}{d x} 1 + \frac{d}{d x} \left(- x\right)\right)}{2} + \frac{\left(1 - x\right) \frac{d}{d x} x}{2}

   Sum rule: differentiate term by term.

5. \frac{x \frac{d}{d x} \left(- x\right)}{2} + \frac{\left(1 - x\right) \frac{d}{d x} x}{2}

   The derivative of a constant is 0.

6. - \frac{x \frac{d}{d x} x}{2} + \frac{\left(1 - x\right) \frac{d}{d x} x}{2}

   Constant multiple rule: pull the constant out.

7. \frac{1}{2} - x

   d/dx of x is 1.

8. f^{(1)}(x) = \frac{1}{2} - x

   That is derivative number 1.

9. \frac{d}{d x} \frac{1}{2} + \frac{d}{d x} \left(- x\right)

   Sum rule: differentiate term by term.

10. \frac{d}{d x} \left(- x\right)

    The derivative of a constant is 0.

11. - \frac{d}{d x} x

    Constant multiple rule: pull the constant out.

12. -1

    d/dx of x is 1.

13. f^{(2)}(x) = -1

    That is derivative number 2.