If y=ln(xa+bx)x,then x3d2ydx2 is equal to

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Question

If $y = l n {(\frac{x}{a + b x})}^{x}$ ,then $x^{3} \frac{d^{2} y}{d x^{2}}$ is equal to

Options:

A .

{(\frac{d y}{d x} + x)}^{2}

B .

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

C .

{(x \frac{d y}{d x} + y)}^{2}

D .

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

Answer: Option D
:
D

∵ y = l n {(\frac{x}{a + b x})}^{x} = x (l n x - l n (a + b x))

o r (\frac{y}{x}) = l n x - l n (a + b x)

Differentiating both sides w.r.t.x,then

\frac{x \frac{d y}{d x} - y .1}{x^{2}} = \frac{1}{x} - \frac{b}{a + b x} = \frac{a}{x (a + b x)} \dots (i)

o r (x \frac{d y}{d x} - y) = (\frac{a x}{a + b x})

Again taking logarithm on both sides, then

l n (x \frac{d y}{d x} - y) = l n (a x) - l n (a + b x)

Differetiating both sides w.r.t.x, then

\frac{x \frac{d^{2} y}{d x^{2}} + \frac{d y}{d x} - \frac{d y}{d x}}{(x \frac{d y}{d x} - y)} = \frac{1}{x} - \frac{b}{a + b x} = \frac{a}{x (a + b x)} = \frac{(x \frac{d y}{d x} - y)}{x^{2}} [F r o m E q . (i)] o r x^{3} \frac{d^{2} y}{d x^{2}} = {(x \frac{d y}{d x} - y)}^{2}

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