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Derivatives — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsDerivatives2 Marks
Question
Differentiate the following functions with respect to x:$\frac{\text{ax}^2+\text{bx}+\text{c}}{\text{px}^2+\text{qx}+\text{r}}$
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Answer

$\frac{\text{d}}{\text{dx}}\Big(\frac{\text{ax}^2+\text{bx}+\text{c}}{\text{px}^2+\text{qx}+\text{r}}\Big)$$=\frac{(\text{px}^2+\text{qx}+\text{r})\frac{\text{d}}{\text{dx}}(\text{ax}^2+\text{bx}+\text{c})-(\text{ax}^2+\text{bx}+\text{c})\frac{\text{d}}{\text{dx}}(\text{px}^2+\text{qx}+\text{r})}{(\text{px}^2+\text{qx}+\text{r})^2}$
$=\frac{(\text{px}^2+\text{qx}+\text{r})(\text{2ax}+\text{b})-(\text{ax}^2+\text{bx}+\text{c})(\text{2px}+\text{q})}{(\text{px}^2+\text{qx}+\text{r})^2}$
$=\frac{2\text{apx}^3+\text{2aqx}^2+\text{2axr}+\text{bpx}^2+\text{bqx}+\text{br}-(\text{2apx}^3+\text{2pbx}^2+\text{2pcx}+\text{qax}^2+\text{bqx}+\text{cq})}{(\text{px}^2+\text{qx}+\text{r})^2}$
$=\frac{\text{2apx}^3-\text{2apx}^3+\text{2aqx}^2+\text{bpx}^2-\text{2pbx}^2-\text{qax}^2+\text{2arx}+\text{bqx}-\text{2pcx}-\text{bqx}+\text{br}-\text{cq}}{(\text{px}^2+\text{qx}+\text{r})^2}$
$=\frac{\text{aqx}^2-\text{bpx}^2+\text{2axr}-\text{2cpx}+\text{br}-\text{cq}}{(\text{px}^2+\text{qx}+\text{r})^2}$
$\frac{\text{x}^2(\text{aq}-\text{bp})+2(\text{ar}-\text{cp})\text{x}+\text{br}-\text{cq}}{(\text{px}^2+\text{qx}+\text{r})^2}$
$\frac{(\text{aq}-\text{bp})\text{x}^2+2(\text{ar}-\text{cp})\text{x}+\text{br}-\text{cq}}{(\text{px}^2+\text{qx}+\text{r})^2}$

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