- WE WANT TO DETERMINE 
THE SECOND DERIVATIVE
OF THE GIVEN FUNCTION,
WHICH MEANS WE NEED TO FIND THE 
DERIVATIVE
OF THE FIRST DERIVATIVE.
SO TO DETERMINE THE FIRST 
DERIVATIVE OF THE GIVEN FUNCTION
WE NEED TO RECOGNIZE THAT 
THIS IS A COMPOSITE FUNCTION.
SO WE'LL HAVE TO APPLY 
THE EXTENDED POWER RULE
THAT INCLUDES THE CHAIN RULE.
SO WE'LL LET THE INTER FUNCTION 
OF 2X SQUARED - 3 = U.
SO NOW WE CAN REWRITE THE 
FUNCTION IN TERMS OF U
AS U TO THE 4th.
SO IF WE LET U = 2X TO THE 2nd 
- 3,
WE KNOW WE'RE GOING TO NEED 
U PRIME
SO LET'S GO AHEAD AND FIND IT 
NOW.
U PRIME IS GOING TO BE EQUAL 
TO 4X.
SO NOW WE HAVE ALL THE 
INFORMATION WE NEED
TO FIND THE FIRST DERIVATIVE 
OF THE FUNCTION.
THE FIRST DERIVATIVE IS GOING TO 
BE EQUAL TO THE DERIVATIVE
OF U TO THE 4th 
WITH RESPECTS TO X.
SO WE'LL HAVE 4U TO THE 3rd x U 
PRIME
OR 4U IS 2X TO THE 2nd MINUS 3 
TO THE 3rd
x U PRIME AND U PRIME IS 4X.
SO TO CLEAN THIS UP, THE FIRST 
DERIVATIVE WOULD BE EQUAL TO 16X
x THE QUANTITY 2X SQUARED - 3 
RAISED TO THE 3rd POWER.
BUT REMEMBER, OUR GOAL IS TO 
DETERMINE THE SECOND DERIVATIVE.
SO WE WANT TO FIND THE 
DERIVATIVE
OF THIS FUNCTION HERE.
AND NOTICE HOW NOW WE HAVE 
A PRODUCT OF TWO FUNCTIONS,
SO WE'LL HAVE TO APPLY 
THE PRODUCT RULE.
SO WE'LL LET THE FIRST FUNCTION 
EQUAL F
AND THE SECOND FUNCTION EQUAL G.
SO LETS START BY WRITING OUT 
THE PRODUCT RULE
AND JUST IN CASE YOU NEED 
THE REVIEW,
IT'S STATED HERE AT THE BOTTOM.
SO THE SECOND DERIVATIVE IS 
GOING TO BE EQUAL
TO THE FIRST FUNCTION OR F
x THE DERIVATIVE OF THE 2nd 
OR THE DERIVATIVE OF G.
AND NOTICE TO FIND THE 
DERIVATIVE OF THIS PART
WE'LL HAVE TO APPLY 
THE EXTENDED POWER RULE,
WHICH INCLUDES THE CHAIN RULE 
AGAIN
PLUS THE SECOND FUNCTION, OR G,
x THE DERIVATIVE OF THE FIRST, 
OR THE DERIVATIVE OF F.
NOW LET'S FOCUS ON DETERMINING 
THE DERIVATIVE HERE
AND THE DERIVATIVE HERE AND THEN 
WE'LL SEE WHAT WE HAVE LEFT.
SO F DOUBLE PRIME OF X = 16X.
NOW x THE DERIVATIVE OF THE 
QUANTITY 2X TO THE 2nd
MINUS 3 TO THE 3rd.
AGAIN, WE HAVE TO APPLY THE 
EXTENDED POWER RULE HERE
WHERE U IS GOING TO BE EQUAL TO 
2X TO THE 2nd - 3.
SO YOU CAN THINK OF THIS AS U TO 
THE 3rd.
SO WE'D HAVE x 3 U TO THE 2nd 
x U PRIME,
WHERE U IS 2X TO THE 2nd - 3 
AND U PRIME WOULD BE 4X.
PLUS THE QUANTITY 2X SQUARED 
- 3 TO THE 3rd
x THE DERIVATIVE OF 16X, 
WHICH WOULD JUST BE 16.
NOW WE'RE GOING TO SIMPLIFY 
THIS FUNCTION
AND THE FIRST THING WE'RE GOING 
TO DO
IS FACTOR OUT THE GREATEST 
COMMON FACTOR OF THIS PRODUCT
AND THIS PRODUCT.
SO NOTICE HOW WE HAVE 
A COMMON FACTOR OF 16
AS WELL AS 2 FACTORS OF 2X 
TO THE 2nd - 3.
SO WE'RE GOING TO FACTOR OUT 16
AS WELL AS 2 FACTORS OF X 
SQUARED - 3.
LET'S SEE WHAT'S LEFT.
HERE WE'D BE LEFT WITH X x 3 
x 4X = 12X SQUARED
PLUS HERE WE'RE FACTORING OUT 
THE 16,
BUT WE'D STILL HAVE 1 FACTOR 
OF 2X SQUARED - 3.
AND THE LAST STEP HERE WOULD BE 
TO COMBINE THE LIKE TERMS.
HERE WE HAVE 12X SQUARED 
AND 2X SQUARED.
SO OUR SECOND DERIVATIVE 
IS GOING TO BE EQUAL TO 16
x THE QUANTITY 2X SQUARED - 3 
TO THE 2nd x 14X SQUARED - 3.
THIS WOULD BE OUR SECOND 
DERIVATIVE FUNCTION.
SO I HOPE YOU FOUND THIS 
HELPFUL.
