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If $y = x - x^{2},$ then the derivative of $y^{2}$ with respect to $x^{2}$ is

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$1 - 2 x$

$2 - 4 x$

$3 x - 2 x^{2}$

$1 - 3 x + 2 x^{2}$

answer is D.

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Detailed Solution

$y = x - x^{2} \Rightarrow \frac{y}{x} = 1 - x^{2}, Also \frac{d y}{d x} = 1 - 2 x$

$Now \frac{d (y^{2})}{d (x^{2})} = \frac{\frac{d (y^{2})}{d x}}{\frac{d (x^{2})}{d x}}$

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

$= 1 - 3 x + 2 x^{2} = 2 x^{2} - 3 x + 1$

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