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in this lesson we're going to focus on

finding the difference quotient of a

function so let's say we have a function

f of X and it's equal to 7x what is the

difference quotient of that function now

the difference quotient is represented

by this formula it's f of X plus h minus

f of X over H now we already have f of X

is 7x what do you think f of X plus h

represents well all you got to do is

replace X with X plus h so it's 7 times

X plus h so this expression becomes 7 X

plus h and then minus f of X which is

already just 7 X divided by H now we'll

need to distribute 7 to X plus h so 7

times X that's going to be 7x and then 7

times H so we're gonna have plus 7 H and

then minus the 7x that was already there

so 7x minus 7x those two will cancel and

so we'll left behind with 7 H divided by

H and so H divided by H is 1 so they

cancel leaving behind 7 and so that's

the answer the difference quotient of 7x

is 7 try this example let's say that f

of X is 5x plus 4 go ahead and find the

difference quotient of 5x plus 4 so

first let's find out what f of x plus h

is going to be so all I need to do is

replace X with X plus h everywhere you

seen X so it's a 5 times X plus h plus 4

so let's replace this with that

expression

and then f of X is just 5 X plus 4 but

we need to put that within the

parentheses so first let's distribute

the 5 so 5 times X is 5x and then we

have 5 times H which is 5 H and then

plus 4 and that will need to distribute

the negative sign to everything inside

the parenthesis so this is going to be

negative 5 X minus 4 so we could cancel

5 X 5 X plus the negative 5 X adds up to

0 we can cancel 4 and negative 4 so

we're left with 5 H divided by H and so

we could cancel H leave it behind the

final answer of 5 so the difference

quotient of 5 X plus 4 is equal to 5 now

here's another one that you could try

let's say that f of X is equal to x

squared determine the difference

quotient of x squared so first let's

determine what f of X plus h is going to

be so all we need to do is replace X

with X plus h so it's going to be X plus

h squared so we're going to have X plus

h squared and f of X is x squared

divided by H now X plus h squared is the

same as X plus h times another X plus h

so in this example we need to foil so

let's multiply X by X that's going to

give us x squared and then we have x

times H so that's simply X H and then H

times X that's another X H and then H

times H is H squared and then we have

negative x squared

so x squared plus negative x squared

adds up to 0 and then X H plus XH that's

2 X H and then we have an H squared

added to it divided by H now once you

get to this part because both terms

contain an H you want to factor out the

GCF the greatest common factor so if we

take out H from the numerator if you

take away H from 2 X H is just going to

be 2x and if you take away H from H

squared there's gonna be one H left over

and so what we can do is cancel these

two ages and so the difference quotient

is going to be 2 X plus h so that's the

difference quotient of x squared it's 2

X plus h now what about this one the

square root of x go ahead and determine

the difference quotient so f of X plus h

is going to be the square root of x plus

h so as always let's start with the

formula now let's replace f of X plus h

with the square root of x plus h and f

of X is simply the square root of x now

what do you think we'll need to do at

this point if you have a radical the

best thing to do is to multiply the top

and the bottom by the conjugate of the

radical the conjugate is going to look

just like this but with the opposite

sign so since we have a negative sign

we're going to use a plus sign and

whatever you do to the numerator of the

fraction you must do to the denominator

of the fraction so that the value of the

fraction remains the same now let's foil

the square root of x plus h times the

square root of x plus h the square roots

will cancel giving us X plus h and then

we have the square root of x plus h

times the square root of x which i'm

simply just going to write it like this

and then negative square root X times

negative square root H I mean square

root X plus h and then we have negative

square root X times positive square root

X that's just negative X so if we

multiply the square root of x times the

square root of x its equal to X for

instance if you multiply the square root

of 4 times the square root of 4 that's

the square root of 16 which is 4 so the

two square roots will cancel leave them

behind the stuff that's on the inside

now and that's not mater we simply have

H times this stuff the square root of x

plus h plus the square root of x so X

and negative x will cancel and these two

terms they're exactly the same but they

have opposite signs so they will cancel

as well so here's what we have left over

all we have is an H on top and then the

same stuff on the bottom

so now we could cancel H H divided by H

is 1 so the final answer is 1 divided by

the square root of x plus h plus the

square root of x so that's the

difference quotient of the square root

of x now the next example we're going to

work on is 1 over X so f of X plus h

it's going to be 1 over X plus h so

let's calculate the difference quotient

so f of X plus h times 1 over X plus h

and then minus f of X or 1 of X now if

you have a complex fraction like this

the best thing to do is to multiply the

top and the bottom by the common

denominator of these two fractions which

will be X and X plus h

now when you multiply 1 over X plus h by

x times X plus h the X plus h terms will

cancel leaving behind 1 times X which is

simply X now when you multiply 1 over X

by this X will cancel leaving behind X

plus h with a negative sign in front of

it so it's gonna be negative X plus h

then everything in the bottom just

rewrite it together so we have an X and

H and an X plus h the order in which you

write it doesn't matter now we do need

to distribute the negative sign so it's

going to be negative X minus H positive

X and negative X will add up to 0 and so

this will leave behind negative H

divided by x times H times X plus h now

let's cancel H so the final answer will

be negative 1 divided by x times X plus

h so that's the difference quotient of 1

over X this is going to be the last

example let's say f of X is 3x squared

plus 4x minus 5 calculate the difference

quotient so let's start with the formula

f of X plus h minus f of X divided by H

so what is f of X plus h in its example

well everywhere you see an X replace it

with X plus h so it's going to be 3

times X plus h squared plus 4 times X

plus h minus 5 so all of this is f of X

H then we need to subtract that by f of

X which is what we have here so let's

put all of that inside of parenthesis

now we need to foil X plus h squared

from a previous example when we multiply

2x plus h by X plus h if you recall we

got x squared plus 2xh plus h squared

but if you want to see it worked out

real quick here it is so it's going to

be x times X which is x squared and then

x times H that's xh h times X is also XH

H times H is H squared then these two

you add together one plus one is two so

you get x squared plus 2xh plus h

squared so now let's distribute the four

this is going to be 4x plus 4-h and then

we have minus 5 and then let's

distribute the negative sign so it's

going to be negative 3x squared minus 4x

but plus 5 all divided by H now let's

distribute the 3 so it's going to be 3x

squared and then we have 3 times 2 XH

which is 6 XH and then plus 3h squared

now we could cancel 4x and negative 4x

and we can cancel negative 5 and 5 so we

have just 4-h and negative 3x squared so

now we can cancel 3x squared and

negative 3x squared and for the terms

that are left over each term contains an

H so for the remaining three terms let's

take out an H so this is going to be 6 X

plus 3 H plus 4 and so we can now cancel

the H outside of that so the final

answer is sick

X plus 4 plus 3h or you could say 6x

plus 3h plus 4 doesn't matter but that's

the difference quotient of 3x squared

plus 4x minus 5