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  • Need solution for RD Sharma maths class 12 chapter 10 Differentiation exercise Multiple choice question 7

Need solution for RD Sharma maths class 12 chapter 10 Differentiation exercise Multiple choice question 7

Answers (1)

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Answer:

        \frac{\log x}{(1+\log x)^{2}}

Hint:

        Differentiate the function w.r.t x

Given:

        x^{y}=e^{x-y}

Solution:  

        x^{y}=e^{x-y}

Taking log on both sides

        \log x^{y}=\log \left(e^{x-y}\right)

        y \log x=(x-y) \log e \quad\left[\because \log m^{n}=n \log m\right]

        \begin{aligned} &y \log x=(x-y) \cdot 1\quad\quad\quad[\because \log e=1] \\\\ &y \log x=x-y \end{aligned}

        \begin{aligned} &x=y+y \log x \\\\ &x=y(1+\log x) \\\\ &y=\frac{x}{1+\log x} \end{aligned}

Differentiating y w.r.t x then

        \frac{d y}{d x}=\frac{d}{d x}\left(\frac{x}{1+\log x}\right)

              =\frac{(1+\log x) \frac{d(x)}{d x}-x \frac{d}{d x}(1+\log x)}{(1+\log x)^{2}} \quad\left[\because \frac{d}{d x}\left(\frac{u}{v}\right)=\frac{v \frac{d u}{d x}-u \frac{d v}{d x}}{v^{2}}\right]

              =\frac{(1+\log x) \cdot 1-x\left\{\frac{d(1)}{d x}+\frac{d(\log x)}{d x}\right\}}{(1+\log x)^{2}}\left[\because \frac{d}{d x}\{f(x)+g(x)\}=\frac{d}{d x} f(x)+\frac{d}{d x} g(x), \frac{d(x)}{d x}=1\right]

              =\frac{1+\log x-x\left(0+\frac{1}{x}\right)}{(1+\log x)^{2}} \quad\left[\because \frac{d}{d x} \log x=\frac{1}{x}, \frac{d}{d x}(\operatorname{con} s \tan t)=0\right]

              =\frac{1+\log x-x \cdot \frac{1}{x}}{(1+\log x)^{2}}=\frac{1+\log x-1}{(1+\log x)^{2}}

              =\frac{\log x}{(1+\log x)^{2}}

 

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