# How do you differentiate #f(x)=xe^(2x)sinx# using the product rule?

Given function:

differentiating above function using product rule as follows

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To differentiate ( f(x) = xe^{2x}\sin(x) ) using the product rule, we apply the rule which states that the derivative of the product of two functions ( u(x) ) and ( v(x) ) is given by ( u'(x)v(x) + u(x)v'(x) ). Here, ( u(x) = x e^{2x} ) and ( v(x) = \sin(x) ).

Applying the product rule:

( f'(x) = u'(x)v(x) + u(x)v'(x) )

( f'(x) = (1 \cdot e^{2x} + x \cdot (2e^{2x}))\sin(x) + x e^{2x} \cdot \cos(x) )

( f'(x) = e^{2x}\sin(x) + 2xe^{2x}\sin(x) + x e^{2x} \cdot \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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