If $y = x - x^{2}$ then the derivative of $y^{2}$ wrt to $x^{2}$ 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 option is B $2 x^{2} - 3 x + 1$
$y = x - x^{2} \Rightarrow y^{2} = {(x - x^{2})}^{2}$
$\frac{d}{d x} (y^{2}) = \frac{d}{d x} ({(x - x^{2})}^{2}) = 2 (x - x^{2}) (1 - 2 x)$
$\frac{d (x^{2})}{d x} = 2 x$
Using $\frac{d (y^{2})}{d (x^{2})} = \frac{d (y^{2})}{d x} \div \frac{d (x^{2})}{d x}$
Therefore derivative of $y^{2}$ w.r.t $x^{2}$ is
$= \frac{2 (x - x^{2}) (1 - 2 x)}{2 x} = {2 x}^{2} - 3 x + 1$

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