Find the difference quotient of f(x). Simply Fully.

Question

Answer 1

f(x+h) = 2(x+h) / (x+h-1)

The NUMERATOR of the difference quotient is:

f(x+h)-f(x)=

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

[2(x+h)(x-1) - 2x(x+h-1) ] / [ (x+h-1)(x-1)]

= [ 2(x^2 -x + xh -h ) -2x(x+h-1)] / [(x+h-1)(x-1)]

= [ 2x^2 -2x + 2xh - 2h -2x^2 - 2xh + 2x] / [(x+h-1)(x-1)]

..... 2x^2,2x and 2xh cancel out...

The Numerator of difference quotient becomes:

[ -2h]/[ (x+h-1)(x-1)]

The denominator of the difference quotient is just h,

which cancels the h in the numerator

the complete difference quotient is:

-2/[(x+h-1)(x-1)]