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Provide solution for RD Sharma maths class 12 chapter Differentiation exercise 10.5 question 34

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Answer: x \cdot \frac{d y}{d x}=2 y

Hint: Differentiate the equation by taking log on both the sides

Given:  x^{16} y^{9}=\left(x^{2}+y\right)^{17}


x^{16} y^{9}=\left(x^{2}+y\right)^{17}

Taking log on both sides,

\begin{aligned} &\log \left(x^{16} y^{9}\right)=\log \left(x^{2}+y\right)^{17} \\\\ &\log \left(x^{16}\right)+\log \left(y^{9}\right)=\log \left(x^{2}+y\right)^{17} \end{aligned}

\frac{d}{d x} 16 \log (x)+\frac{d}{d x} 9 \log (y)=\frac{d}{d x}\left(17 \log \left(x^{2}+y\right)\right)

\frac{16}{x}+\frac{9}{y} \cdot \frac{d y}{d x}=17 \cdot \frac{1}{x+y} \cdot\left[\left(x^{2}+y\right) \frac{d y}{d x}\right]

\frac{16}{x}-\frac{34}{x^{2}+y}=\left(\frac{17}{x^{2}+y}-\frac{9}{y}\right) \frac{d y}{d x}

\begin{aligned} &\frac{16 x^{2}+16 y-34 x}{x\left(x^{2}+y\right)}=\left(\frac{17 y-9 x^{2}-9 y}{\left(x^{2}+y\right) y} \cdot \frac{d y}{d x}\right. \\\\ &\frac{-18 x^{2}+16 y}{x\left(x^{2}+y\right)}=\frac{-9 x^{2}+8 y}{\left(x^{2}+y\right) y} \cdot \frac{d y}{d x} \end{aligned}

 x \cdot \frac{d y}{d x}=2 y

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