- 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.