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Provide solution for RD Sharma maths class 12 chapter Differentiation exercise 10.5 question 62

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Answer: \frac{d y}{d x}=y \times\left(\log y+\frac{x}{y}\right)

Hint: To solve this equation we use log on both side

Given: x^{y}-y^{x}=a^{b}



Taking log on both side

        \begin{aligned} &\log x^{y}=\log z \\\\ &y \log x=\log z \\\\ &\frac{d y}{d x} \log x+\frac{y}{x}=\frac{1}{z} \frac{d z}{d x} \end{aligned}

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

        \begin{aligned} &y^{x}=p \\\\ &\log y^{x}=\log p \\\\ &x \log y=\log p \end{aligned}

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

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