- 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.