# If f(x)=log(1-x)/(1+x).then what will be the value of f'(0)?

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To find the value of ( f'(0) ), we first need to differentiate the function ( f(x) ) with respect to ( x ) using the quotient rule. Then, we substitute ( x = 0 ) into the derivative expression.

Given ( f(x) = \frac{{\log(1-x)}}{{1+x}} ),

Using the quotient rule, the derivative of ( f(x) ) with respect to ( x ) is:

[ f'(x) = \frac{{(1+x) \cdot \frac{{d}}{{dx}}(\log(1-x)) - \log(1-x) \cdot \frac{{d}}{{dx}}(1+x)}}{{(1+x)^2}} ]

[ f'(x) = \frac{{(1+x) \cdot \left(\frac{{-1}}{{1-x}}\right) - \log(1-x) \cdot 1}}{{(1+x)^2}} ]

[ f'(x) = \frac{{-1-x - \log(1-x)}}{{(1+x)(1-x)}} ]

Now, substitute ( x = 0 ) into ( f'(x) ):

[ f'(0) = \frac{{-1 - 0 - \log(1-0)}}{{(1+0)(1-0)}} ]

[ f'(0) = \frac{{-1 - \log(1)}}{{1}} ]

[ f'(0) = \frac{{-1 - 0}}{{1}} ]

[ f'(0) = -1 ]

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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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