Differentiate the following functions w.r.t. x. : x x - Vidyadip

Question

Differentiate the following functions w.r.t. x. : $\frac{x}{\log x}$

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Answer

Let $y=\frac{x}{\log x}$
Differentiating w.r.t. $x$, we get
$
\begin{aligned}
\frac{\mathrm{d} y}{\mathrm{~d} x} & =\frac{\mathrm{d}}{\mathrm{d} x}\left(\frac{x}{\log x}\right)_{\text { }} \\
& =\frac{\log x \frac{\mathrm{d}}{\mathrm{d} x}(x)-x \frac{\mathrm{d}}{\mathrm{d} x}(\log x)}{(\log x)^2} \\
& =\frac{\log x(1)-x\left(\frac{1}{x}\right)}{(\log x)^2} \\
& =\frac{\log x-1}{(\log x)^2}
\end{aligned}
$

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