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

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

Answers (1)

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}

Solution:  

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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