How do you differentiate #y=e^x/(1+x) #?

Answer 1

#-e^x/(1+x)^2+e^x/(1+x)#

There are a few ways to do this, but the easiest is probably to separate the numerator and denominator into two different fractions. #e^x*(1/(1+x))#
You can now use the product rule: #d/dxf(x)*g(x)=f'(x)*g(x)+f(x)*g'(x)#
The derivative of #y=e^x# is #dy/dx=e^x# and the derivative of #y=1/(1+x)# is #dy/dx=-1/(1+x)^2# (You can get this using the power rule.)

From here, just put in the values:

#e^x*(-1/(1+x)^2)+(1/(1+x))*e^x#

This is pretty messy and doesn't really simplify

#-e^x/(1+x)^2+e^x/(1+x)#
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Answer 2

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:

  1. Identify ( u ) and ( v ):

    • Let ( u = e^x ) and ( v = 1 + x ).
  2. 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).
  3. Apply the quotient rule:

    • ( y' = \frac{(e^x)(1+x) - (e^x)(1)}{(1+x)^2} ).
  4. 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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Answer from HIX Tutor

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.

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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