Let F(x)=x+3/x+1. The difference quotient for f(x) at x=a is defined f(a+h)-f(a)/h. For the given f, fully simplify the difference quotient

F(x) = (x+3)/(x+1).

The given
definition of difference coefficient d/dx f(x) =
{f(a+h)-fa)}/h.

To find the difference coefficient for the
given function  F(x):

Diference coefficient F(x) at a is
given by:

{F(a+h)-F(a)}/h = {(a+3+h)/(a+1+h) -
(a+1)/(a+1)}/h.

F(x) = (x+3)/(x+1) = (x+1+2)/(x+1) =
1+2/(x+1)

Therefore { F(x+h)-F(x)}/h  =  {1 -1 +(2/(x+1+h)
- 2/(x+1)}/h

={2 (x+1) -
2(x+1+h)}/[(x+1+h)(x+1)h]

= {2x+2
-2x-2-2h}/[(x+1+h)(x+1)h]

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

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