First slide

Derivatives

Question

$if, y = x \log (\frac{x}{a + b x}) then x^{3} \frac{d^{2} y}{d x^{2}} =$

Moderate

A

$x \frac{d y}{d x} - y$
B

${(x \frac{d y}{d x} - y)}^{2}$
C

$y \frac{d y}{d x} - x$
D

${(y \frac{d y}{d x} - x)}^{2}$

Solution

$\frac{y}{x} = \log x - \log (a + b x)$
$\begin{array}{l} \frac{x \frac{d y}{d x} - y}{x^{2}} = \frac{1}{x} - \frac{b}{a + b x} = \frac{a}{x (a + b x)} \\ ∴ x \frac{d y}{d x} - y = \frac{a x}{a + b x} \dots \dots . . . (1) \end{array}$
Diff again w..r.t ‘x’ we get
$\begin{array}{l} x \frac{d^{2} y}{d x^{2}} + \frac{d y}{d x} - \frac{d y}{d x} = \frac{(a + b x) a - a x \cdot b}{(a + b x)^{2}} \\ x \frac{d^{2} y}{d x^{2}} = \frac{a^{2}}{(a + b x)^{2}} \\ x^{3} \frac{d^{2} y}{d x^{2}} = \frac{a^{2} x^{2}}{(a + b x)^{2}} = {(x \frac{d y}{d x} - y)}^{2} (b y e q (1)) \end{array}$

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