The derivative of log_{10} x with respect to | vedclass

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Question

(B) Let $u = \log_{10} x = \frac{\ln x}{\ln 10}$ and $v = \log_x 10 = \frac{\ln 10}{\ln x}$.We need to find $\frac{du}{dv}$.Note that $v = \frac{(\ln

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$-(\log_{10} x)^2$

Vedclass · Answer

(B) Let $u = \log_{10} x = \frac{\ln x}{\ln 10}$ and $v = \log_x 10 = \frac{\ln 10}{\ln x}$.We need to find $\frac{du}{dv}$.Note that $v = \frac{(\ln 10)^2}{\ln x} \cdot \frac{1}{\ln 10} = \frac{1}{u}$.Thus, $v = u^{-1}$.Then $\frac{dv}{du} = -u^{-2} = -\frac{1}{u^2}$.Therefore, $\frac{du}{dv} = \frac{1}{dv/du} = -u^2$.Since $u = \log_{10} x$, we have $\frac{du}{dv} = -(\log_{10} x)^2$.

The derivative of $\log_{10} x$ with respect to $\log_x 10$ is

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