Differentiate the following functions with respect to x: _x3 - Vidyadip

Question

Differentiate the following functions with respect to x:
$\log_\text{x}3$

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Answer

Let, $\text{y}=\log_\text{x}3$
$\Rightarrow\ \text{y}=\frac{\log3}{\log\text{x}}\ \Big[\because\ \log_\text{a}\text{b}=\frac{\log\text{b}}{\log\text{a}}\Big]$
Differentiate it with respect to x we get,
$\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\Big(\frac{\log3}{\log\text{x}}\Big)$
$=\log3\frac{\text{d}}{\text{dx}}(\log\text{x})^{-1}$
$=\log3\times\Big[-1(\log\text{x})^{-2}\Big]\frac{\text{d}}{\text{dx}}(\log\text{x})$
[Using chain rule]
$=-\frac{\log 3}{(\log\text{x})^2}\times\frac{1}{\text{x}}$
$=-\Big(\frac{\log 3}{\log\text{x}}\Big)^2\times\frac{1}{\text{x}}\times\frac{1}{\log3}$
$=-\frac{1}{\text{x}\log3(\log_3\text{x})^2} \Big[\because \frac{\log\text{b}}{\log\text{a}}=\log_\text{a}\text{b}\Big]$
So,
$\frac{\text{d}}{\text{dx}}(\log_\text{x}3)=-\frac{1}{\text{x}\log3(\log_3\text{x})^2}$

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