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

1. Apply the quotient rule, which is:

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

   $f{\left(x \right)} = x^{2} + 5$ and $g{\left(x \right)} = 2 - x$.

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

   1. Differentiate $x^{2} + 5$ term by term:

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

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

      The result is: $2 x$

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

   1. Differentiate $2 - x$ term by term:

      1. The derivative of the constant $2$ 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: $-1$

   Now plug in to the quotient rule:

   $\frac{x^{2} + 2 x \left(2 - x\right) + 5}{\left(2 - x\right)^{2}}$

2. Now simplify:

   $$\frac{x^{2} - 2 x \left(x - 2\right) + 5}{\left(x - 2\right)^{2}}$$

The answer is:

$$\frac{x^{2} - 2 x \left(x - 2\right) + 5}{\left(x - 2\right)^{2}}$$