Right let's take a look at question number
seven we are to find F prime if F is the absolute
value of three X plus two okay so there two
possible solutions you can have for this the
derivative of this function so the first one
I'm going to be using the differentiation
rule DDX of the absolute value you equals
you over the absolute value you the you the
X okay this is one possibly of finding the
derivative of this function okay so solution
one were going to be using this formula right
here we also going to be applying the chain
rule okay this is a composition of two functions
is a composition of the absolute value function
and the linear function okay so we have all
F F of G of X equals the absolute value of
three X plus two okay and we know what the
chain rule is right so one of find the derivative
of F of G of X so F of G of X prime is equal
to F prime of G of X the derivative of the
outer function and I evaluated at the inner
function times the derivative of the inner
function 
so we are going to apply the chain rule here
so let's see what the have we have all the
compose this composite function take out F
F of X is the absolute value X that is the
outer function now the inner function is G
of X G of X is of three X plus two okay now
is going to proceed to find the derivative
of these two functions now the derivative
of F F prime of X is X over the absolute value
of X the derivative of G G prime of X is three
so what we going to do is find F prime of
X and I waited at G so we just simply take
G and plug it into the X is the F prime of
X okay and then multiply it by G prime of
X so F prime of X is simply going to be F
prime of G of X is instead of X divided by
the absolute value X within a have three X
is to taking the place of X so have three
X plus two divided by the absolute value three
X plus two using the chain rule formula we
are to multiply this result by the derivative
of the inner function G prime of X okay so
of G prime of X is three so just multiply
this by three so that's your answer okay we
can simplify into F prime of X equals nine
X plus six divided by the absolute value of
three X plus two so there is your first solution
now the second solution that we going to generate
involves expressing this function is piecewise
be find function okay on this result is also
equally correct so solution to solution to
we're going to write the absolute value function
as follows this is going to be we know that
F of X which equals the absolute value three
X this to can be written as plus or minus
three X plus two the sign change happens when
the argument of the absolute value function
changes sign okay so we just have to so three
X was to equals zero so this four X and that
will tell you where you sign change happens
okay so we subtract two and divide by three
we have X equals negative two thirds prime
so now all function is negative three X plus
two when X is less than negative two thirds
and three X plus two when X is greater than
or equal to two negative two thirds okay so
now this is our piece wise find function now
what were going to do is find the derivative
of this piecewise be find function so F prime
of X is simply going to be the derivative
of the function to the left which is negative
three X plus two the derivative of that is
negative T three when X is less than negative
to third and this on the bottom is three when
X is going to then or equal to negative to
third okay so this is another possible solution
for the derivative of the absolute value of
three X plus two okay so any of these solutions
you will get for credit you have to find the
derivative of these absolute value function
thanks so much for taking the time to watch
this presentation really appreciated feel
free to subscribe to our channel for updates
to other cool tutorials such as this and also
feel free to post a comments in the comments
section to let us know what you think about
this lesson more clips can be, mathgotserved.com
thanks again for watching and have a wonderful
day
