If $y=\log ^n x$, where $\log ^n$ means $\log _e \log _e \log _e \ldots$ (repeated $n$ times), then $x \log x \log ^2 x \log ^3 x \ldots . . \log ^{n-1} x \log ^n x \frac{d y}{d x}$ is equal to

Question

If $y=\log ^n x$, where $\log ^n$ means $\log _e \log _e \log _e \ldots$ (repeated $n$ times), then $x \log x \log ^2 x \log ^3 x \ldots . . \log ^{n-1} x \log ^n x \frac{d y}{d x}$ is equal to, $\log x$, $x$, 1, $\log ^{\mathrm{n}} x$

MARKS App · Accepted Answer

Hint : $\because \frac{d y}{d x}=\frac{1}{x \log ^{n-1} x \quad \log ^{n-2} x \ldots . \log x}$

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