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

The following is the product rule: if your function is the result of two functions being combined,

subsequently, the derivative is

Thus, your derivative as a whole is:

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To differentiate ( f(x) = e^{-x} \sin(x) ) using the product rule, you apply the formula:

[ (uv)' = u'v + uv' ]

where ( u = e^{-x} ) and ( v = \sin(x) ).

Then, take the derivatives of ( u ) and ( v ):

[ u' = -e^{-x} ] [ v' = \cos(x) ]

Now, apply the product rule:

[ f'(x) = (e^{-x})(\sin(x))' + (e^{-x})'(\sin(x)) ]

[ = (-e^{-x} \sin(x)) + (e^{-x} \cos(x)) ]

[ = e^{-x} (\cos(x) - \sin(x)) ]

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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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