How do you differentiate #y=e^x/(1+x) #?
From here, just put in the values:
This is pretty messy and doesn't really simplify
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To differentiate ( y = \frac{e^x}{1+x} ), you can use the quotient rule, which states that if you have a function ( \frac{u}{v} ), then its derivative is given by ( \frac{u'v  uv'}{v^2} ). Here's how to apply it:

Identify ( u ) and ( v ):
 Let ( u = e^x ) and ( v = 1 + x ).

Find the derivatives of ( u ) and ( v ):
 ( u' = e^x ) (since the derivative of ( e^x ) is itself).
 ( v' = 1 ) (since the derivative of ( 1 + x ) is 1).

Apply the quotient rule:
 ( y' = \frac{(e^x)(1+x)  (e^x)(1)}{(1+x)^2} ).

Simplify the expression:
 ( y' = \frac{e^x(1+x)  e^x}{(1+x)^2} ).
 ( y' = \frac{e^x + xe^x  e^x}{(1+x)^2} ).
 ( y' = \frac{xe^x}{(1+x)^2} ).
So, the derivative of ( y = \frac{e^x}{1+x} ) with respect to ( x ) is ( \frac{xe^x}{(1+x)^2} ).
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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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