If $y = x - x^{2}$ , then the derivative 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 option is C $2 x^{2} - 3 x + 1$
Given, $y = x - x^{2}$
On differentiating w.r.t. $x$ , we get
$\frac{d y}{d x} = 1 - 2 x$
Now, $\frac{d (y^{2})}{d (x^{2})} = \frac{\frac{d (y^{2})}{d x}}{d (x^{2})} d (x)$
$= \frac{2 y \frac{d y}{d x}}{2 x} = 2 y \frac{(1 - 2 x)}{2 x}$
$= \frac{2 (x - x^{2}) (1 - 2 x)}{2 x}$
$= (1 - x) (1 - 2 x)$
$= 1 - 3 x + 2 x^{2}$
$= 2 x^{2} - 3 x + 1$ .

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