First slide

Derivatives

Question

$If y = (\log \underset{}{\underset{}{x}})^{x} then \frac{d y}{d x} =$

Moderate

A

${(\log x)}^{x} [\log (\log x) + \frac{1}{\log x}]$
B

${(\log x)}^{x} [\log (\log x) - \frac{1}{\log x}]$
C

$- {(\log x)}^{x} [\log (\log x) + \frac{1}{\log x}]$
D

$- {(\log x)}^{x} [\log (\log x) - \frac{1}{\log x}]$

Solution

$\begin{array}{l} y = (\log x)^{x} \\ \log y = \log (\log x)^{x} \\ \frac{1}{y} \frac{d y}{d x} = \log (\log x) + x \frac{1}{\log x} \frac{d}{d x} \log x \\ = \log (\log x) + \frac{x}{\log x} \frac{1}{x} \\ \frac{d y}{d x} = y [\log (\log x) + \frac{1}{\log x}] \end{array}$

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