Find \(\frac{dy}{dx}\), if \(y = (\log x)^x\).
Q.9 Find \(\frac{dy}{dx}\), if \(y = (\log x)^x\).
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Answer: \(\frac{dy}{dx} = (\log x)^x \left[\log(\log x) + \frac{1}{\log x}\right]\)
Taking logarithm on both sides: \(\log y = x \log(\log x)\).
Differentiating w.r.t. \(x\): \(\frac{1}{y}\frac{dy}{dx} = x \cdot \frac{1}{\log x} \cdot \frac{1}{x} + \log(\log x) \cdot 1 = \frac{1}{\log x} + \log(\log x)\).
\(\therefore \frac{dy}{dx} = (\log x)^x \left[\log(\log x) + \frac{1}{\log x}\right]\).
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· 29 Oct 2026
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