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need solution for RD Sharma maths class 12 chapter Differentiation exercise 10.2 question 73

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Hint: you must know the rules of solving derivatives.

Given:x y=4

Prove: x\left(\frac{d y}{d x}+y^{2}\right)=3 y


\begin{gathered} x y=4 \\\\ y=\frac{4}{x} \end{gathered} 

Differentiate with respect to x,

\begin{aligned} &\frac{d y}{d x}=4 \frac{d}{d x}\left(x^{-1}\right) \\\\ &\frac{d y}{d x}=4(-1) \times\left(x^{-1-1}\right) \end{aligned}

\frac{d y}{d x}=-\frac{4}{x^{2}}

\frac{d y}{d x}=-\frac{y^{2}}{4}           ( multiplying by 4 in num. & den.)


\begin{aligned} &4 \frac{d y}{d x}=-y^{2} \\\\ &4 \frac{d y}{d x}=3 y^{2}-4 y^{2} \\\\ &4 \frac{d y}{d x}+4 y^{2}=3 y^{2} \\\\ &4\left(\frac{d y}{d x}+4 y^{2}\right)=3 y^{2} \end{aligned}

Dividing both sides with x,

\begin{aligned} &\frac{4}{x}\left(\frac{d y}{d x}+y^{2}\right)=\frac{3 y^{2}}{x} \\\\ &y\left(\frac{d y}{d x}+y^{2}\right)=\frac{3 y^{2}}{x} \\\\ &x\left(\frac{d y}{d x}+y^{2}\right)=\frac{3 y^{2}}{y} \\ &x\left(\frac{d y}{d x}+y^{2}\right)=3 y \end{aligned}

∴ Proved


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