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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
