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

1 - 2 x

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2 - 4 x

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3 x - 2 x^{2}

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1 - 3 x + 2 x^{2}

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Solution

The correct option is D $1 - 3 x + 2 x^{2}$
Let $u = y^{2}$ and $v = x^{2}$
$∴ \frac{d u}{d x} = \frac{d}{d x} y^{2} = (\frac{d}{d y} y^{2}) (\frac{d y}{d x})$
$= 2 y (1 - 2 x) = 2 (x - x^{2}) (1 - 2 x) = 2 x (1 - x) (1 - 2 x)$ -- (1)
and $\frac{d v}{d x} = 2 x$ -- (2)
Hence, $\frac{d u}{d v} = \frac{\frac{d u}{d x}}{\frac{d v}{d x}} = \frac{2 x (1 - x) (1 - 2 x)}{2 x}$ $[$ from (1) and (2) $]$
$= (1 - x) (1 - 2 x) = 1 - 3 x + 2 x^{2}$

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