$$K = \frac{-(\sigma^2 + 3\sigma + 2)}{\sigma^2 - 8\sigma +15}$$
How is this differentiated with respect to $\sigma$?
The answer is
$$\dfrac{dK}{d\sigma} = \frac{11\sigma^2 - 26\sigma -61}{(\sigma^2 - 8\sigma + 15)^2}$$
And the quotient rule is a very long process, I get the feeling there was a short method.