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

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

Answers (1)

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}

Solution:  

        x^{y}=z

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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