The differential coefficient of log _{10} x w | vedclass

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Question

(B) Let $y = \log _{10} x$ and $z = \log _{x} 10$. We know that $\log _{a} b = \frac{\ln b}{\ln a}$. So,$y = \frac{\ln x}{\ln 10}$ and $z = \frac{\ln

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$-\left(\log _{10} x\right)^{2}$

Vedclass · Answer

(B) Let $y = \log _{10} x$ and $z = \log _{x} 10$. We know that $\log _{a} b = \frac{\ln b}{\ln a}$. So,$y = \frac{\ln x}{\ln 10}$ and $z = \frac{\ln 10}{\ln x}$. Thus,$y = \frac{1}{z}$. We need to find the derivative of $y$ with respect to $z$,which is $\frac{dy}{dz}$. Since $y = z^{-1}$,differentiating with respect to $z$ gives $\frac{dy}{dz} = -z^{-2} = -\frac{1}{z^{2}}$. Substituting $z = \frac{1}{y}$ back into the expression,we get $\frac{dy}{dz} = -y^{2}$. Since $y = \log _{10} x$,the result is $-\left(\log _{10} x\right)^{2}$.

The differential coefficient of $\log _{10} x$ with respect to $\log _{x} 10$ is

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