If $y = x - x^{2}$ , then the derivatives of $y^{2}$ w.r.t. $x^{2}$ is

2 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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None of these

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Solution

The correct options are
B $2 x^{2} - 3 x + 1$
C $2 x^{2} + 3 x - 1$
Consider the given equation.
$y = x - x^{2}$

On squaring both sides, we get

$y^{2} = x^{2} + x^{4} - 2 x^{3}$ …….. (1)

On differentiating with respect to $x^{2}$ , we get

$\frac{d y^{2}}{d x^{2}} = \frac{d}{d x^{2}} ⎛ ⎜ ⎝ x^{2} + {(x^{2})}^{2} - 2 {(x^{2})}^{\frac{3}{2}} ⎞ ⎟ ⎠$

$\frac{d y^{2}}{d x^{2}} = 1 + 2 x^{2} - 2 (\frac{3}{2} x)$

$\frac{d y^{2}}{d x^{2}} = 2 x^{2} - 3 x + 1$

So, this is the derivative of given expression.

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