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
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