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APPLICATION OF DERIVATIVES — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAPPLICATION OF DERIVATIVES5 Marks
Question
$\text{If y =} \log\bigg(\frac{\text{x}}{\text{a + bx}}\bigg)^{\text{x}}, \text{prove that x}^{3} \frac{\text{d}^{2}\text{y}}{\text{dx}^{2}} = \bigg(\text{x}\frac{\text{dy}}{\text{dx}} - \text{y}\bigg)^{2}.$
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Answer

$\frac{\text{y}}{\text{x}} = [\log\text{x} - \log\text{(a + b x)}]$
$\Rightarrow \frac{\text{x}\frac{\text{dy}}{\text{dx}} - \text{y}}{\text{x}^{2}} = \frac{1}{\text{x}} - \frac{\text{b}}{\text{a + bx}}$
$\Rightarrow \text{x}\frac{\text{dy}}{\text{dx}} - \text{y} = \frac{\text{ax}}{\text{a + bx}} \dots\dots\dots\dots\dots\dots\text{(i)}$
Differentiating again,
$\text{x}\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}} = \frac{\text{a}^{2}}{(\text{a + bx)}^{2}}$
$\text{x}^{3}. \frac{\text{d}^{2}\text{y}}{\text{dx}^{2}} = \bigg(\frac{\text{ax}}{\text{a + bx}}\bigg)^{2} = \bigg(\text{x}\frac{\text{dy}}{\text{dx}}-y\bigg)^{2} \text{(using (i))}$

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