How do you differentiate #f(x)= e^x/(xe^(x) x )# using the quotient rule?
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To differentiate the function ( f(x) = \frac{e^x}{xe^x  x} ) using the quotient rule:

Identify ( u(x) ) and ( v(x) ) as follows: ( u(x) = e^x ) and ( v(x) = xe^x  x ).

Apply the quotient rule formula: [ \frac{d}{dx}\left(\frac{u(x)}{v(x)}\right) = \frac{u'(x)v(x)  v'(x)u(x)}{[v(x)]^2} ]

Find the derivatives of ( u(x) ) and ( v(x) ): ( u'(x) = e^x ) and ( v'(x) = e^x + xe^x  1 ).

Substitute the derivatives and the original functions into the quotient rule formula: [ f'(x) = \frac{(e^x)(xe^x  x)  (e^x + xe^x  1)(e^x)}{(xe^x  x)^2} ]

Simplify the expression if necessary.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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