The sides of two squares are $x$ and $y$ respectively, such that $y = x + x^{2}$ . The rate of change of area of second square with respect to area of first square is ________.

x^{2} + 3 x - 1

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2 x^{2} - 3 x + 1

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2 x^{2} + 3 x + 1

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1 + 2 x

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Solution

The correct option is C $2 x^{2} + 3 x + 1$
$x$ and $y$ are the sides of two squares respectively, such that $y = x + x^{2}$
Let $A_{1}$ and $A_{2}$ be the areas of first and second squares respectively
Then, $A_{1} = x^{2}$ and $A_{2} = y^{2}$
$\Rightarrow \frac{d A_{2}}{d A_{1}} = \frac{2 y \frac{d y}{d t}}{2 x \frac{d x}{d t}}$
$= \frac{y}{x} \frac{d y}{d x}$
Since $y = x + x^{2} \Rightarrow \frac{d y}{d x} = 1 + 2 x$
$∴ \frac{d A_{2}}{d A_{1}} = \frac{y}{x} \times (1 + 2 x)$
$= 2 x^{2} + 3 x + 1$ .

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The sides of two squares are x and y respectively, such that y=x+x2. The rate of change of area of second square with respect to area of first square is ________.

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