- WELCOME TO ANOTHER VIDEO 
ON HIGHER-ORDER DERIVATIVES.
I WOULD RECOMMEND WATCHING 
THE FIRST VIDEO I MADE
ON HIGHER-ORDER DERIVATIVES
BECAUSE IT DOES GO INTO 
MUCH MORE DETAIL
ABOUT THE USES OF HIGHER-ORDER 
DERIVATIVES.
THIS VIDEO JUST PROVIDES SOME 
EXTRA EXAMPLES
OF FINDING HIGHER-ORDER 
DERIVATIVES
OF TRIGONOMETRIC FUNCTIONS.
LET'S FIRST REVIEW THE DIFFERENT 
NOTATION THAT WE CAN USE
WHEN WE'RE TALKING ABOUT 
THE FIRST, SECOND, THIRD, FOURTH
AND NTH DERIVATIVES
AND THESE ARE REFERRED TO AS 
HIGHER-ORDER DERIVATIVES.
LET'S GO AHEAD AND LOOK 
AT A COUPLE EXAMPLES.
HERE WE HAVE F OF X = SINE X 
AND WE WANT TO FIND THE FIRST,
SECOND, THIRD, AND FOURTH 
DERIVATIVES OF THIS FUNCTION.
WELL THE FIRST DERIVATIVE 
WOULD BE COSINE X,
AND THE SECOND DERIVATIVE WOULD 
BE THE DERIVATIVE
OF THE FIRST DERIVATIVE,
AND THE DERIVATIVE OF COSINE X 
IS EQUAL TO -SINE X.
WELL THE THIRD DERIVATIVE 
WOULD BE THE DERIVATIVE
OF THE SECOND DERIVATIVE.
THE DERIVATIVE OF -SINE X WOULD 
BE -COSINE X
AND THEN LASTLY, 
THE FOURTH DERIVATIVE,
SINCE THE DERIVATIVE OF COSINE X 
IS -SINE X
AND WE HAVE A NEGATIVE HERE 
WE'RE GOING TO HAVE
A - -SINE X 
WHICH IS EQUAL TO SINE X.
WHAT WE'LL NOTICE HERE IS THAT 
THE FOURTH DERIVATIVE
IN THE ORIGINAL FUNCTION 
ARE THE SAME
AND LET'S JUST TAKE A LOOK 
AT ONE MORE EXAMPLE.
HERE WE WANT TO FIND THE FIRST 
AND SECOND DERIVATIVE
OF F OF X = SECANT X.
WELL THE FIRST DERIVATIVE 
IS A REVIEW.
THE DERIVATIVE OF SECANT X 
= SECANT X TANGENT X.
NOW OUR SECOND DERIVATIVE IS 
GOING TO TAKE A LITTLE MORE WORK
BECAUSE THIS IS ACTUALLY 
A PRODUCT.
SO WE'RE GOING TO HAVE TO USE 
THE PRODUCT RULE
WHERE THIS IS OUR FIRST FUNCTION 
F
AND THIS IS OUR SECOND FUNCTION 
G.
SO LET'S GO AHEAD 
AND WRITE OUT THE PRODUCT RULE.
THE SECOND DERIVATIVE 
IS EQUAL TO THE FIRST FUNCTION F
x THE DERIVATIVE OF THE SECOND 
FUNCTION G
+ THE SECOND FUNCTION
x THE DERIVATIVE OF THE FIRST 
FUNCTION.
OKAY, LET'S GO AHEAD 
AND FIND THESE DERIVATIVES NOW
AND THEN SEE IF WE CAN SIMPLIFY 
THIS.
THE DERIVATIVE OF TANGENT X 
IS EQUAL TO SECANT SQUARED X
AND THE DERIVATIVE OF SECANT X 
IS EQUAL TO SECANT X TANGENT X.
NOW YOU'LL NOTICE 
THESE TWO PRODUCTS
DO HAVE A COMMON FACTOR 
OF SECANT X.
LET'S GO AHEAD 
AND FACTOR THAT OUT.
WE'D BE LEFT WITH THE QUANTITY 
SECANT SQUARED X
+ TANGENT SQUARED X.
NOW ONE OF THE CHALLENGES 
OF FINDING DERIVATIVES
OF TRIGONOMETRIC FUNCTIONS 
IS MAKING THE FORM MATCH
WHAT YOU MAY FIND 
IN THE BACK OF THE BOOK
OR IN YOUR ONLINE HOMEWORK.
WHAT THING THAT I SEE HERE 
THEY MIGHT TRY TO DO
IS WRITE THIS IN TERMS OF 
ONE TRIG FUNCTION.
REMEMBER THAT TANGENT SQUARED X 
+ 1
IS EQUAL TO SECANT SQUARED X.
SO WHAT WE COULD DO IS REPLACE 
TANGENT SQUARED X
WITH SECANT SQUARED X - 1
AND WHAT THAT WOULD DO IS HAVE 
THE SECOND DERIVATIVE
ALL IN TERMS OF SECANT 
AND A CONSTANT.
SO LET'S GO AHEAD AND DO THAT.
LET'S REPLACE TANGENT SQUARED X 
WITH SECANT SQUARED X - 1
AND NOW YOU'LL NOTICE WE DO HAVE 
TWO COMMON FACTORS HERE
INSIDE THE PARENTHESES.
SO LET'S GO AHEAD AND SIMPLIFY 
THIS ONE MORE TIME.
SO WE'D HAVE SECANT SQUARED X 
- 1 INSIDE THE PARENTHESES.
SO AGAIN THIS VIDEO JUST 
PROVIDED A COUPLE EXTRA EXAMPLES
OF HIGHER-ORDER DERIVATIVES
DEALING WITH 
TRIGONOMETRIC FUNCTIONS.
I HOPE YOU FOUND THE EXAMPLES 
HELPFUL.
