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

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f(x) = x+3/ x+1

f(a) = a+3/a+1

f(a+h) = a+h+3/a+h+1

f(a+h) - f(a) /h = [(a+h+3)/(a+h+1) - (a+3/(a+1)]/h

= (a+h+3)(a+1) - (a+3)(a+h+1)/h(a+1)(a+h+1)

= (a^2 + 4a + ah + h + 3 - (a^2 + ah + 4a + 3h + 3)/h(a+1)(a+h+1)

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

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

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)

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