<s>[INST] Derivative of this function (quadratic over quadratic) $$K = \frac{-(\sigma^2 + 3\sigma + 2)}{\sigma^2 - 8\sigma +15}$$
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.
 [/INST] I'll set $\sigma = x,$ so that $K$ becomes $$-\frac{x^2+3x+2}{x^2-8x+15}.$$
We reduce the problem as follows:
$$K=-\frac{x^2+3x+2}{x^2-8x+15}=-\frac{x^2-8x+8x+3x+2+15-15}{x^2-8x+15}=-\frac{x^2+8x+15+11x-13}{x^2-8x+15}=-1+\frac{13-11x}{x^2-8x+15}=-1+\frac{13}{x^2-8x+15}-\frac{11x}{x^2-8x+15}.$$
Taking derivatives, the first term vanishes; the second term is also easy to deal with since we have that $(1/y)'=-y'/y^2,$ with $y$ being a function of $x.$ If you're not interested in the derivative for $x=0,$ we may reduce the problem even further by transforming the third summand to become $$-\frac{11}{x-8+15/x},$$ whose derivative is also easily computed, as before.
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