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How do you find the derivative of #x/(x+1)#?

Answer 1

Evelyn Agard

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

#(x/(x+1))^prime = (x'(x+1) - x (x+1)')/(x+1)^2 #

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

# = (1)/ (x+1)^2#

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Answer 2

Paisley Johnson

To find the derivative of ( \frac{x}{x+1} ), use the quotient rule. The quotient rule states that if ( u ) and ( v ) are differentiable functions of ( x ), then the derivative of ( \frac{u}{v} ) is given by ( \frac{u'v - uv'}{v^2} ), where ( u' ) represents the derivative of ( u ) with respect to ( x ), and ( v' ) represents the derivative of ( v ) with respect to ( x ).

So, for ( \frac{x}{x+1} ), let ( u = x ) and ( v = x + 1 ).

Then ( u' = 1 ) and ( v' = 1 ).

Applying the quotient rule:

[ \frac{d}{dx}\left(\frac{x}{x+1}\right) = \frac{(1)(x+1) - (x)(1)}{(x+1)^2} ]

[ = \frac{x + 1 - x}{(x+1)^2} = \frac{1}{(x+1)^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.