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If $\sqrt{\frac{y}{x}}+4 \sqrt{\frac{x}{y}}=4$, then $\frac{d y}{d x}=$
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The correct answer is:
$4$
Given, $\sqrt{\frac{y}{x}}+4 \sqrt{\frac{x}{y}}=4$
On squaring both sides, we get
$\begin{array}{rlrl} & & \frac{y}{x}+\frac{16 x}{y}+8 & =16 \Rightarrow \frac{y}{x}+16 \frac{x}{y}=8 \\ \Rightarrow & y^2+16 x^2 & =8 x y\end{array}$
On differentiating both sides w.r.t. $x$, we get
$\begin{aligned} 2 y \frac{d y}{d x}+32 x & =8 y+8 x \frac{d y}{d x} \\ \Rightarrow \quad & (4 x-y) \frac{d y}{d x}=4(4 x-y) \Rightarrow \frac{d y}{d x}=4\end{aligned}$
Hence, option (d) is correct.
On squaring both sides, we get
$\begin{array}{rlrl} & & \frac{y}{x}+\frac{16 x}{y}+8 & =16 \Rightarrow \frac{y}{x}+16 \frac{x}{y}=8 \\ \Rightarrow & y^2+16 x^2 & =8 x y\end{array}$
On differentiating both sides w.r.t. $x$, we get
$\begin{aligned} 2 y \frac{d y}{d x}+32 x & =8 y+8 x \frac{d y}{d x} \\ \Rightarrow \quad & (4 x-y) \frac{d y}{d x}=4(4 x-y) \Rightarrow \frac{d y}{d x}=4\end{aligned}$
Hence, option (d) is correct.
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