TO DETERMINE THE DERIVATIVE 
OF F OF X
= SQUARE ROOT OF X CUBED - 4,
WE NEED TO RECOGNIZE THIS 
AS A COMPOSITE FUNCTION
WHERE THERE IS AN INNER FUNCTION 
AND AN OUTER FUNCTION,
AND THEREFORE WE WILL HAVE 
TO APPLY THE CHAIN RULE.
AND THE CHAIN RULE TELLS US 
TO DETERMINE THE DERIVATIVE
OF A COMPOSITE FUNCTION
WE NEED TO DETERMINE THE 
DERIVATIVE OF THE OUTER FUNCTION
AND THEN MULTIPLY IT 
BY THE DERIVATIVE
OF THE INNER FUNCTION.
SO WHEN WE HAVE 
A COMPOSITE FUNCTION
WE WANT TO LET U 
EQUAL THE INNER FUNCTION.
SO IN THIS CASE U WOULD BE X 
TO THE 3rd - 4.
AND LET'S GO AHEAD 
AND WRITE THAT OUT OVER HERE.
U = X CUBED - 4,
WHICH MEANS WE COULD WRITE THIS 
AS THE SQUARE ROOT OF U
AND WE KNOW BY NOW 
IF WE HAVE A SQUARE ROOT
WE WANT TO REWRITE THIS 
USING A RATIONAL EXPONENT
WHERE THE EXPONENT HERE IS ONE 
AND THE INDEX IS TWO.
SO THIS IS EQUAL TO U 
TO THE 1/2 POWER.
ONCE WE HAVE OUR FUNCTION 
WRITTEN IN TERMS OF U
IT'S A VERY STRAIGHT FORWARD 
PROCESS TO APPLY THE CHAIN RULE.
WE NEED TO FIND THE DERIVATIVE 
OF U TO THE 1/2
AS WE NORMALLY DO
AND THEN MULTIPLY IT 
BY THE DERIVATIVE OF U.
AND USUALLY ONCE YOU LEARN 
THE CHAIN RULE
ALL OF YOUR DIFFERENTIATION 
RULES ARE REWRITTEN
SO THAT THEY CONTAIN A U
AS WE SEE HERE FOR THE EXTENDED 
OR GENERAL POWER RULE.
AND NOTICE HOW THE ONLY 
DIFFERENCE HERE
IS THAT WE HAVE A U HERE 
INSTEAD OF X,
AND THEN WE HAVE THE DERIVATIVE 
IN TERMS OF U x U PRIME,
WHICH AGAIN IS JUST 
THE CHAIN RULE
TELLING US TO FIND THE 
DERIVATIVE OF THE OUTER FUNCTION
AND THEN MULTIPLY IT 
BY THE DERIVATIVE
OF THE INNER FUNCTION.
SO F PRIME OF X IS GOING TO BE 
EQUAL TO THE DERIVATIVE OF U
TO THE 1/2.
THAT WOULD BE 1/2 U 
TO THE 1/2 - 1 x U PRIME.
AND NOW WE NEED TO REWRITE 
THIS DERIVATIVE IN TERMS OF X
RATHER THAN U.
SO WE'LL REPLACE U WITH X 
TO THE 3rd -4
AND WE'LL REPLACE U PRIME 
WITH THE DERIVATIVE OF U
WHICH WOULD BE 3X SQUARED.
SO WE'RE GOING TO HAVE 1/2 x U, 
WHICH IS X TO THE 3RD - 4,
TO THE POWER OF 1/2 - 1.
THAT'S -1/2 x U PRIME 
AND U PRIME IS 3X SQUARED.
SO THIS IS OUR DERIVATIVE 
FUNCTION,
BUT NOW WE DO HAVE 
TO SIMPLIFY THIS.
SO IF WE THINK OF THIS AS BEING 
OVER ONE WE CAN MORE THIS DOWN
SO THAT OUR EXPONENT 
WOULD BE POSITIVE.
LET'S ALSO PUT THIS OVER ONE
SO WE KNOW WHAT PART 
IS IN THE NUMERATOR
AND WHAT'S IN THE DENOMINATOR.
SO OUR DERIVATIVE FUNCTION 
IS GOING TO BE EQUAL
TO THE FRACTION
WHERE THE NUMERATOR 
WOULD BE 3X SQUARED
AND THE DENOMINATOR WOULD BE 2
x (X TO THE 3rd - 4 
TO THE POSITIVE 1/2 POWER).
AND IF WE WANTED TO 
WE COULD WRITE THIS
AS THE SQUARE ROOT 
OF X CUBED -4.
SO LET'S GO AHEAD AND DO THAT.
WE HAVE 3X SQUARED,
AND OUR DENOMINATOR IS 2 
x (SQUARE ROOT OF X CUBED - 4).
SO AS YOU CAN SEE,
AS LONG AS WE'VE IDENTIFIED 
THE INNER FUNCTION AS U,
WE WRITE THE FUNCTION 
IN TERMS OF U,
APPLYING THE CHAIN RULE 
IS PRETTY STRAIGHT FORWARD.
WE FIND THE DERIVATIVE OF U 
AND THEN MULTIPLY IT BY U PRIME.
I HOPE YOU HAVE FOUND THIS 
HELPFUL.
