Derivative of (1-cos(x))/x^2

1. Apply the quotient rule, which is:

   $$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$$

   $f{\left(x \right)} = 1 - \cos{\left(x \right)}$ and $g{\left(x \right)} = x^{2}$.

   To find $\frac{d}{d x} f{\left(x \right)}$:

   1. Differentiate $1 - \cos{\left(x \right)}$ term by term:

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

      2. The derivative of a constant times a function is the constant times the derivative of the function.

         1. The derivative of cosine is negative sine:

            $$\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$$

         So, the result is: $\sin{\left(x \right)}$

      The result is: $\sin{\left(x \right)}$

   To find $\frac{d}{d x} g{\left(x \right)}$:

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

   Now plug in to the quotient rule:

   $\frac{x^{2} \sin{\left(x \right)} - 2 x \left(1 - \cos{\left(x \right)}\right)}{x^{4}}$

2. Now simplify:

   $$\frac{x \sin{\left(x \right)} + 2 \cos{\left(x \right)} - 2}{x^{3}}$$

The answer is:

$$\frac{x \sin{\left(x \right)} + 2 \cos{\left(x \right)} - 2}{x^{3}}$$