What is the first (x/(1-x))^' ?

The first (x/(1-x))^' is 1/((1-x)^2)

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(x/(1-x))^'

(\frac{x}{1 - x})^{^{'}}

\frac{1}{( 1 - x ) ^{2}}

Solution steps

(\frac{x}{1 - x})^{'}

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{x ^{^{'}} ( 1 - x ) - ( 1 - x ) ^{^{'}} x}{( 1 - x ) ^{2}}

x^{'} = 1

(1 - x)^{'} = - 1

= \frac{1 \cdot ( 1 - x ) - ( - 1 ) x}{( 1 - x ) ^{2}}

Simplify

\frac{1 \cdot ( 1 - x ) - ( - 1 ) x}{( 1 - x ) ^{2}} : \frac{1}{( 1 - x ) ^{2}}

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