Differentiate the following functions with respect to x: px^2+qx+… - Vidyadip

Question

Differentiate the following functions with respect to x:$\frac{\text{px}^2+\text{qx}+\text{r}}{\text{ax}+\text{b}}$

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Answer

We have,$\frac{\text{d}}{\text{dx}}\Big(\frac{\text{px}^2+\text{qx}+\text{r}}{\text{ax}+\text{b}}\Big)$
Using quotient rule, we get
$\frac{(\text{ax}+\text{b})\frac{\text{d}}{\text{dx}}(\text{px}^2+\text{qx}+\text{r})-(\text{px}^2+\text{qx}+\text{r})\frac{\text{d}}{\text{dx}}(\text{ax}+\text{b})}{(\text{ax}+\text{b})^2}$
$=\frac{(\text{ax}+\text{b})(\text{2px}+\text{q})-(\text{px}^2\text{qx}+\text{r})\text{a}}{(\text{ax}+\text{b})^2}$
$=\frac{\text{2apx}^2+\text{2pbx}+\text{aqx}+\text{bq}-\text{apx}^2-\text{aqx}-\text{ar}}{(\text{ax}+\text{b})^2}$
$=\frac{\text{apx}^2+\text{2pbx}+\text{bq}-\text{ar}}{(\text{ax}+\text{b})^2}$

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