Differential coefficient of log_10x with respect to log_x10 is :

The correct answer is Option (3) → \(-\displaystyle\frac{1}{(\log_{x}10)^2}\)

Let

    \(y = \log_{10}x,\quad z = \log_{x}10\)

By the change of base rule, \(\log_{x}10\) and \(\log_{10}x\) are reciprocals:

    \( \displaystyle z = \log_{x}10 = \frac{1}{\log_{10}x} = \frac{1}{y} \)

So

    \( \displaystyle y = \frac{1}{z} \)

    \( \displaystyle \frac{dy}{dz} = -\frac{1}{z^2} \)

    \( \displaystyle = -\frac{1}{(\log_{x}10)^2} \)