The product and quotient rule

Question

f(x) = 1/x  + 1/1-x  (in final answer, use a common denominator and simplify numerator)

Answer 1

f(x) = 1/x + 1/(1-x) = (1-x+x) / (x-x²) = 1 / (x-x²)

So, f'(x) = [(x-x²)(0) - (1-2x)(1)] / (x-x²)² = (2x-1) / [x²(1-x)²]

Answer 2

1/x + 1/(1-x)

=(1-x)/x(1-x) + x/x(1-x)

= (1-x +x)/x(1-x)

=1/(x-x^2)

a somewhat surprising simplification for the numerator

as a rough check, try letting x=a simple integer, such as x=2

then 1/x+1/(1-x)= 1/2 + 1/-1= 1/2-1 = - 1/2

compare that to 1/(x-x^2) =1/(2-4) = 1/-2 = -1/2

It seems to work, which gives major confidence in the simplified fraction

try x=1, and you get undefined both ways, division by zero