Differential coefficient of log<sub>10</sub> x with respect to log<sub>x</sub> 10 is
Let \(u = \log_{10}x = \frac{\ln x}{\ln 10}\) and \(v = \log_x 10 = \frac{\ln 10}{\ln x}\).
\(\frac{du}{dx} = \frac{1}{x\ln 10}\) and \(\frac{dv}{dx} = -\frac{\ln 10}{x(\ln x)^2}\).
\(\frac{du}{dv} = \frac{du/dx}{dv/dx} = \frac{1/(x\ln 10)}{-\ln 10/(x(\ln x)^2)} = -\frac{(\ln x)^2}{(\ln 10)^2}\)
(Or directly: \(u = \frac{1}{v}\), so \(\frac{du}{dv} = -\frac{1}{v^2} = -u^2\).)
Answer: (A)
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