# How do you differentiate #f(x)=xlnx-x#?

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To differentiate ( f(x) = x \ln x - x ), you can use the product rule and the differentiation of natural logarithm.

The product rule states that if you have two functions ( u(x) ) and ( v(x) ), then the derivative of their product is given by ( (u \cdot v)' = u'v + uv' ).

In this case, ( u(x) = x ) and ( v(x) = \ln x ).

The derivative of ( x ) with respect to ( x ) is ( 1 ) and the derivative of ( \ln x ) is ( \frac{1}{x} ).

Applying the product rule, we have:

[ f'(x) = (x \cdot \ln x)' = (1 \cdot \ln x + x \cdot \frac{1}{x}) = \ln x + 1 ]

So, the derivative of ( f(x) = x \ln x - x ) is ( f'(x) = \ln 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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