Expert: Sherman D. - 1/24/2005

Question
Find the value of   [f(x+h)-f(x)]/ for this funtion. Be sure to simplify:

F(x)= 1/x+3

Answer
if what you really meant to type was

[f(x + h) - f(x)]/h

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

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

so the substitution would be

[((1/((x + h) + 3)) - (1/(x + 3))]/h

Multiply everything by

(x + 3)(x + h + 3)

[((x + 3) - (x + h + 3))/((x + 3)(x + h + 3))] / h

[((x + 3 - x - h - 3)/((x + 3)(x + h + 3))] / h

This leaves you with

[(-h)/((x + 3)(x + h + 3))]/h

[(-h)/((x + 3)(x + h + 3))]/(h/1)

[(-h)/((x + 3)(x + h + 3))] * (1/h)

the h's cancel, leaving you with

(-1)/((x + 3)(x + h + 3))

becomes

-(1/(x^2 + xh + 3x + 3x + 3h + 9))

becomes

ANS : -(1/(x^2 + (x + 3)h + 6x + 9))