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

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To differentiate ( f(x) = e^x \sin(x) ) using the product rule, you apply the rule which states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.

Using this rule, we have: [ f'(x) = (e^x \cdot \sin(x))' = (e^x)' \cdot \sin(x) + e^x \cdot (\sin(x))' ]

Now, we differentiate each term separately: [ (e^x)' = e^x ] [ (\sin(x))' = \cos(x) ]

Substituting these derivatives back into the expression, we get: [ f'(x) = e^x \cdot \sin(x) + e^x \cdot \cos(x) ]

So, the derivative of ( f(x) = e^x \sin(x) ) with respect to ( x ) using the product rule is ( f'(x) = e^x \sin(x) + e^x \cos(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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