How do you differentiate #f(x)= ( 2x^2+ 3x ) / (  e^x + 2 ) # using the quotient rule?
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To differentiate ( f(x) = \frac{2x^2 + 3x}{e^x + 2} ) using the quotient rule, follow these steps:

Identify ( u(x) ) and ( v(x) ): ( u(x) = 2x^2 + 3x ) ( v(x) = e^x + 2 )

Find ( u'(x) ) and ( v'(x) ): ( u'(x) = 4x + 3 ) ( v'(x) = e^x )

Apply the quotient rule: ( f'(x) = \frac{v(x)u'(x)  u(x)v'(x)}{[v(x)]^2} )

Substitute the values: ( f'(x) = \frac{(e^x + 2)(4x + 3)  (2x^2 + 3x)(e^x)}{(e^x + 2)^2} )

Simplify the expression as needed.
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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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