# How do you find the derivative of #y = x^2 e^(-x)#?

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To find the derivative of ( y = x^2 e^{-x} ), you can use the product rule. The product rule states that if ( y = u \cdot v ), where ( u ) and ( v ) are functions of ( x ), then the derivative of ( y ) with respect to ( x ) is given by:

[ \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} ]

Here, ( u = x^2 ) and ( v = e^{-x} ). The derivatives of ( u ) and ( v ) with respect to ( x ) are:

[ \frac{du}{dx} = 2x ] [ \frac{dv}{dx} = -e^{-x} ]

Now, applying the product rule:

[ \frac{dy}{dx} = x^2 (-e^{-x}) + e^{-x} (2x) ] [ \frac{dy}{dx} = -x^2 e^{-x} + 2x e^{-x} ] [ \frac{dy}{dx} = e^{-x} (2x - x^2) ]

So, the derivative of ( y = x^2 e^{-x} ) with respect to ( x ) is ( e^{-x} (2x - x^2) ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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