# How do you differentiate #f(x)=x^2e^(x-1)# using the product rule?

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To differentiate ( f(x) = x^2e^{x-1} ) using the product rule, follow these steps:

- Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = e^{x-1} ).
- Apply the product rule: ( f'(x) = u'(x)v(x) + u(x)v'(x) ).
- Find the derivatives of ( u(x) ) and ( v(x) ): ( u'(x) = 2x ) (derivative of ( x^2 )) and ( v'(x) = e^{x-1} ) (derivative of ( e^{x-1} )).
- Substitute these derivatives into the product rule formula: ( f'(x) = (2x)(e^{x-1}) + (x^2)(e^{x-1}) ).

So, the derivative of ( f(x) = x^2e^{x-1} ) with respect to ( x ) is: [ f'(x) = 2xe^{x-1} + x^2e^{x-1} ]

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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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