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

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To differentiate ( f(x) = 2x \ln(x) ) using the product rule, you apply the formula ( (uv)' = u'v + uv' ), where ( u = 2x ) and ( v = \ln(x) ).

( u' = 2 ) (derivative of ( 2x ) with respect to ( x ))

( v' = \frac{1}{x} ) (derivative of ( \ln(x) ) with respect to ( x ))

Now apply the product rule:

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

( = (2)(\ln(x)) + (2x)\left(\frac{1}{x}\right) )

( = 2\ln(x) + 2 )

So, ( f'(x) = 2\ln(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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