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(C) Given $y=x \log \left(\frac{1}{a x}+\frac{1}{a}\right) = x \log \left(\frac{1+x}{ax}\right)$.First,find the first derivative $\frac{dy}{dx}$:$\fra

Vedclass · Accepted Answer

$-1$

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(C) Given $y=x \log \left(\frac{1}{a x}+\frac{1}{a}\right) = x \log \left(\frac{1+x}{ax}\right)$.First,find the first derivative $\frac{dy}{dx}$:$\frac{dy}{dx} = \log \left(\frac{1+x}{ax}\right) + x \cdot \frac{ax}{1+x} \cdot \frac{d}{dx} \left(\frac{1+x}{ax}\right)$$= \log \left(\frac{1+x}{ax}\right) + \frac{ax^2}{1+x} \cdot \left(\frac{ax - a(1+x)}{(ax)^2}\right)$$= \log \left(\frac{1+x}{ax}\right) + \frac{ax^2}{1+x} \cdot \left(\frac{-a}{a^2x^2}\right) = \log \left(\frac{1+x}{ax}\right) - \frac{1}{1+x}$.Now,find the second derivative $\frac{d^2y}{dx^2}$:$\frac{d^2y}{dx^2} = \frac{d}{dx} \left(\log \left(\frac{1+x}{ax}\right) - \frac{1}{1+x}\right)$$= \frac{ax}{1+x} \cdot \left(\frac{-1}{ax^2}\right) + \frac{1}{(1+x)^2} = -\frac{1}{x(1+x)} + \frac{1}{(1+x)^2} = \frac{-(1+x) + x}{x(1+x)^2} = \frac{-1}{x(1+x)^2}$.Substitute these into the expression $x(x+1) \frac{d^2 y}{d x^2}+x \frac{d y}{d x}-y$:$= x(x+1) \left(\frac{-1}{x(1+x)^2}\right) + x \left(\log \left(\frac{1+x}{ax}\right) - \frac{1}{1+x}\right) - x \log \left(\frac{1+x}{ax}\right)$$= -\frac{1}{1+x} + x \log \left(\frac{1+x}{ax}\right) - \frac{x}{1+x} - x \log \left(\frac{1+x}{ax}\right)$$= -\left(\frac{1+x}{1+x}\right) = -1$.

If $y=x \log \left(\frac{1}{a x}+\frac{1}{a}\right)$,then $x(x+1) \frac{d^2 y}{d x^2}+x \frac{d y}{d x}-y=$

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