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explain solution RD Sharma class 12 chapter Differentiation exercise 10.2 question 12 maths

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Answer: \frac{-1}{x \log 3\left(\log _{8} x\right)^{2}}

Hint:  You must know the rules of solving derivative of logarithm function

Given: \log _{x} 3

Solution:

Let  y=\log _{x} 3

y=\frac{\log 3}{\log x}                            \left[\therefore \log _{a} b=\frac{\log b}{\log a}\right]

Differentiating with respect to x

\begin{aligned} &\frac{d y}{d x}=\frac{d}{d x}\left(\frac{\log 3}{\log x}\right) \\ &\frac{d y}{d x}=\log 3 \frac{d}{d x}(\log x)^{-1} \end{aligned}                      [  using chain rule]

\begin{aligned} &\frac{d y}{d x}=\log 3 \times\left[-1(\log x)^{-2}\right] \frac{d}{d x}(\log x) \\ &\frac{d y}{d x}=\frac{-\log 3}{(\log x)^{2}} \times \frac{1}{x} \end{aligned}

\begin{aligned} &\frac{d y}{d x}=-\left(\frac{\log 3}{\log x}\right)^{2} \times \frac{1}{x} \times \frac{1}{\log 3} \\ &\frac{d y}{d x}=\frac{-1}{x \log 3\left(\log _{3} x\right)^{2}} \end{aligned}                    \left[\therefore \frac{\log b}{\log a}=\log _{a} b\right]

 

 

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