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Explain solution RD Sharma class 12 chapter Differentiation exercise 10.5 question 40 maths

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Answer: \frac{d y}{d x}=\frac{(1+\log y)^{2}}{\log y}

Hint:  To solve this we differentiate method

Given: y^{x}=e^{y-x}


        \frac{d}{d x} x \log y=(y-x) \log e=\frac{d}{d x}(y-x)

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

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

        \begin{aligned} &\frac{d y}{d x}=\frac{y(1+\log y)}{y-x} \\\\ &\frac{d y}{d x}=\frac{y(1+\log y)}{x \cdot \log y} \\\\ &\frac{d y}{d x}=\frac{(1+\log y)^{2}}{\log y} \end{aligned}


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