Question
Differentiate the following functions with respect to x:
$\frac{\text{ax}+\text{b}}{\text{px}^2+\text{qx}+\text{r}}$
$\frac{\text{ax}+\text{b}}{\text{px}^2+\text{qx}+\text{r}}$
$\frac{\text{d}}{\text{dx}}\Big(\frac{\text{ax}+\text{b}}{\text{px}^2+\text{qx}+\text{r}}\Big)$
$=\frac{(\text{px}^2+\text{qx}+\text{r})\frac{\text{d}}{\text{dx}}(\text{ax}+\text{b})-(\text{ax}+\text{b})\frac{\text{d}}{\text{dx}}(\text{px}^2+\text{qx}+\text{r})}{(\text{px}^2+\text{qx}+\text{r})^2}$
$=\frac{(\text{px}^2+\text{qx}+\text{r})(\text{a})-(\text{ax}+\text{b})(\text{2xp}+\text{q})}{(\text{px}^2+\text{qx}+\text{r})^2}$
$=\frac{(\text{apx}^2+\text{aqx}+\text{ar})-(\text{2apx}^2+\text{aqx}+\text{2bpq}+\text{bq})}{(\text{px}^2+\text{qx}+\text{r})^2}$
$=\frac{-(\text{apx}^2+\text{2bpx}+\text{bq}-\text{ar})}{(\text{px}^2+\text{qx}+\text{r})^2}$
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$\frac{\text{x}+\cos\text{x}}{\tan\text{x}}$