If y = x - x^2,then the rate of change of y^2 | vedclass

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(C) Given $y = x - x^2$. We need to find the rate of change of $y^2$ with respect to $x^2$. Let $v = y^2 = (x - x^2)^2 = x^2 + x^4 - 2x^3$. Let $u = x

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$3$

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(C) Given $y = x - x^2$. We need to find the rate of change of $y^2$ with respect to $x^2$. Let $v = y^2 = (x - x^2)^2 = x^2 + x^4 - 2x^3$. Let $u = x^2$. We want to calculate $\frac{dv}{du} = \frac{dv/dx}{du/dx}$. First,differentiate $v$ with respect to $x$: $\frac{dv}{dx} = \frac{d}{dx}(x^2 + x^4 - 2x^3) = 2x + 4x^3 - 6x^2$. Next,differentiate $u$ with respect to $x$: $\frac{du}{dx} = \frac{d}{dx}(x^2) = 2x$. Now,$\frac{dv}{du} = \frac{2x + 4x^3 - 6x^2}{2x} = 1 + 2x^2 - 3x$. At $x = 2$,the rate of change is $1 + 2(2)^2 - 3(2) = 1 + 8 - 6 = 3$.

If $y = x - x^2$,then the rate of change of $y^2$ with respect to $x^2$ at $x = 2$ is

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