- WE WANT TO DETERMINE 
THE DERIVATIVE
OF THE GIVEN FUNCTION.
NOTICE HOW WE HAVE A QUOTIENT 
RAISED TO THE 3rd POWER.
SO WE HAVE A COMPOSITE FUNCTION 
AS WELL AS A QUOTIENT.
SO TO DETERMINE
WHETHER WE SHOULD APPLY 
THE CHAIN RULE FIRST
OR THE QUOTIENT RULE FIRST,
WE NEED TO RECOGNIZE THAT THIS 
QUOTIENT IS THE INNER FUNCTION
OF A CUBING FUNCTION.
SO WHEN WE HAVE 
THE COMPOSITE FUNCTION,
WE NEED TO LET U 
= THE INNER FUNCTION.
SO IN THIS CASE THIS 
ENTIRE QUOTIENT WOULD BE U.
SO NOW WE CAN WRITE F IN TERMS 
OF U AS JUST U TO THE 3rd.
AND AGAIN WE CAN DO THIS
BECAUSE WE'RE LETTING U EQUAL 
THE QUOTIENT X SQUARED + 2
DIVIDED BY X - 8.
SO NOW WE CAN APPLY 
THE EXTENDED POWER RULE
OR THE POWER RULE WITH THE 
CHAIN RULE AS WE SEE HERE
TO DETERMINE THIS DERIVATIVE.
SO F PRIME OF X 
IS GOING TO BE EQUAL TO
THE DERIVATIVE OF U CUBED 
WITH RESPECTS TO U
WOULD BE 3U SQUARED x U PRIME
WHERE U PRIME IS DERIVATIVE OF U 
WITH RESPECTS TO X.
SO THIS IS OUR DERIVATIVE
BUT OF COURSE WE NEED TO HAVE 
OUR DERIVATIVE IN TERMS OF X.
SO WE COULD SUBSTITUTE 
THIS QUOTIENT FOR U,
BUT WE STILL NEED TO DETERMINE 
U PRIME.
SO LET'S GO OVER HERE ON THE 
SIDE AND DETERMINE U PRIME.
SINCE U IS A QUOTIENT, WE'LL 
HAVE TO APPLY THE QUOTIENT RULE
SO WE'LL LET THE NUMERATOR EQUAL 
F AND THE DENOMINATOR EQUAL G.
SO TO DETERMINE U PRIME,
WE'LL START 
WITH THE DENOMINATOR
AND HERE'S THE QUOTIENT RULE
JUST IN CASE WE NEED 
TO REFERENCE IT.
THE DENOMINATOR 
IS JUST GOING TO BE G SQUARED
SO WE'LL HAVE THE QUANTITY 
X - 8 SQUARED
AND THEN WE'LL HAVE 
THE DENOMINATOR
x THE DERIVATIVE 
OF THE NUMERATOR
MINUS THE NUMERATOR x THE 
DERIVATIVE OF THE DENOMINATOR.
SO AGAIN OUR DENOMINATOR x 
THE DERIVATIVE OF OUR NUMERATOR
WHICH WOULD BE 2X, 
- OUR NUMERATOR
x THE DERIVATIVE 
OF THE DENOMINATOR
WHICH WOULD JUST BE 1,
WELL LET'S SIMPLIFY THIS.
WE HAVE U PRIME 
IS GOING TO BE EQUAL TO--
THE DENOMINATOR STAYS THE SAME,
AND THE NUMERATOR IS GOING TO BE 
2X SQUARED - 16X - X SQUARED - 2
AND NOTICE HOW WE HAVE 
2 LIKE TERMS HERE.
SO FOR U PRIME OUR NUMERATOR IS 
GOING TO BE X SQUARED - 16X - 2.
NOW LET'S GO BACK OVER 
TO THE DERIVATIVE OF F,
WE CAN NOW PERFORM 
THE SUBSTITUTIONS FOR U
AS WELL AS U PRIME.
SO WE'LL HAVE 3 x U SQUARED 
AND U IS THIS QUOTIENT HERE
SO WE'LL HAVE X SQUARED + 2/X 
- 8 RAISED TO THE 2nd x U PRIME
WHICH IS HERE IN GREEN.
SO WE'LL HAVE X SQUARED - 16X 
- 2/THE QUANTITY X - 8 SQUARED.
NOW LET'S GO AHEAD AND MULTIPLY 
THESE FRACTIONS TOGETHER.
THIS FRACTION HERE 
IS BEING SQUARED
SO WE HAVE 2 FACTORS OF X - 8 
HERE AND WE HAVE 2 MORE HERE.
SO WE HAVE A TOTAL OF 4 FACTORS 
OF THE QUANTITY X - 8
AND THEN THE NUMERATOR WOULD BE 
3 x 2 FACTORS OF X SQUARED + 2
AND JUST ONE FACTOR OF OUR 
TRINOMIAL X SQUARED - 16X - 2.
SO THIS ONE WAS KIND OF FUN
BUT THIS WOULD BE OUR 
DERIVATIVE FUNCTION.
I HOPE YOU FOUND THIS HELPFUL.
