Quotients????

[ejames2]

Let f(x)= 7 over 4x-3.

Write the difference quotient f(x+h)-f(x) over h?

 

[soroban]

Hello, ejames2!

You need to do the alegebra . . . carefully!

Let \(\displaystyle f(x)\:=\:\frac{7}{4x\,-\,3}\)
Write the difference quotient: \(\displaystyle \:\frac{f(x\,+\,h)\,-\,f(x)}{h}\)

There are three steps to the Difference Quotient:
. . (1) Find \(\displaystyle f(x+h)\) . . . and simplify
. . (2) Subtract \(\displaystyle f(x)\) . . . and simplify
. . (3) Divide by \(\displaystyle h\) . . . and simplify


We have: \(\displaystyle \:f(x)\:=\:\frac{7}{4x\,-\,3}\)

\(\displaystyle (1) \;f(x+h)\;=\;\frac{7}{4(x+h)\,-\,3} \;=\;\frac{7}{4x\,+\,4h\,-\,3}\)


\(\displaystyle (2)\;f(x+h)\,-\,f(x) \;=\;\frac{7}{4x\,+\,4h\,-\,3}\,-\,\frac{7}{4x\,-\,3}\)

. . .Get a common denominator and subtract: \(\displaystyle \;\frac{7(4x\,-\,3)\,-\,7(4x\,+\,4h\,-\,3)}{(4x\,+\,4h\,-\,3)(4x\,-\,3)}\)

. . .\(\displaystyle = \;\frac{28x\,-\,21\,-\,28x\,-\,28h\,+\,21}{(4x\,+\,4h\,-\,3)(4x\,-\,3)} \;=\;\frac{-28h}{(4x\,+\,4h\,-\,3)(4x\,-\,3)}\)


\(\displaystyle (3)\;\frac{f(x+h)\,-\,f(x)}{h}\;=\;\frac{-28h}{h(4x\,+\,4h\,-\,3)(4x\,-\,3)} \;=\;\L\fbox{\frac{-28}{(4x\,+\,4h\,-\,3)(4x\,-\,3)}}\)

 

[opusman]

first take x+h and sub into function, then subtract off f(x)

7/[4(x+h)-3] - 7/[4x-3] / h

clear your complex fraction by multiplying numerator and denominator by
[4x + 4h -3][4x -3] leaving you with

7[4x-3] - 7[4x+4h-3] / h[4x-3][4x+4h-3]

distribute the numerator parenthesis, terms will cancel leaving you with 1 term in numerator involving an h, you can cancel the h

-28/[4x+4h-3][4x-3]

this is usually connected with a limit as h approaches 0, definition of a derivative
Hopefully this helps.[/tex]