– IF F OF X EQUALS 3X SQUARED
NATURAL LOG 4X, WE WANT TO FIND THE FIRST,
SECOND, AND THIRD DERIVATIVE FUNCTIONS. THE FIRST THING WE SHOULD
RECOGNIZE IS THAT THE GIVEN FUNCTION
IS A PRODUCT OF TWO FUNCTIONS, SINCE WE HAVE 3X SQUARED x
NATURAL LOG 4X. SO TO FIND OUR DERIVATIVES WE’LL
HAVE TO APPLY THE PRODUCT RULE OF DIFFERENTIATION GIVEN HERE. OR IF WE WANT TO FIND THE
DERIVATIVE OF F x G WITH RESPECT TO X THIS WILL EQUAL F x G PRIME
+ G x F PRIME, OR WE CAN SAY THE DERIVATIVE
IS EQUAL TO THE FIRST FUNCTION x THE DERIVATIVE OF THE SECOND
FUNCTION + THE SECOND FUNCTION x THE DERIVATIVE
OF THE FIRST FUNCTION. JUST LOOKING AT THE GIVEN
FUNCTION, WE’LL LET FUNCTION F
=3X SQUARED AND WE’LL LET FUNCTION G=
NATURAL LOG 4X. NOTICE THAT FUNCTION G
IS A COMPOSITE FUNCTION, SO WHEN FINDING G PRIME WE’LL
HAVE TO APPLY THE CHAIN RULE. SO LETS START BY WRITING OUT
THE PRODUCT RULE TO FIND THE FIRST DERIVATIVE. WE’LL HAVE F PRIME OF X IS EQUAL
TO F x G PRIME + G x F PRIME, WHERE THE FIRST FUNCTION x THE DERIVATIVE
OF THE SECOND FUNCTION + THE SECOND FUNCTION x THE DERIVATIVE OF THE FIRST
FUNCTION. NOTICE HOW WE HAVEN’T FOUND ANY
DERIVATIVES YET. WE WROTE OUT THE PRODUCT RULE, SO NOW WE’LL GO BACK AND FIND
THE DERIVATIVE HERE AND HERE. AND THEN FIND THE PRODUCT AND
HOPEFULLY SIMPLIFY. SO NOW WE HAVE F PRIME OF X
=3X SQUARED. NOW TO FIND THE DERIVATIVE
OF NATURAL LOG 4X, WE NEED TO RECOGNIZE THAT THIS
IS A COMPOSITE FUNCTION SO WE’LL LET THE INTER FUNCTION
EQUAL U. SO IN THIS CASE U=4X. SO TO FIND THE DERIVATIVE OF
NATURAL LOG U WITH RESPECT TO X, THIS WILL BE EQUAL TO 1/U
x U PRIME. SO THE DERIVATIVE WOULD BE 1/4X
x THE DERIVATIVE OF 4X, WHICH WOULD BE 4 + NATURAL LOG 4X x THE
DERIVATIVE OF 3X SQUARED. SO WE’RE GOING TO MULTIPLY IT
BY THE EXPONENT, THAT WOULD GIVE US 6
SUBTRACT 1 FROM THE EXPONENT SO OUR DERIVATIVE IS 6X. NOW LET’S SIMPLIFY. NOTICE THIS IS 4/1,
SO 4/4 SIMPLIFIES TO 1 AND WE HAVE 3X SQUARED x 1/X. SO ONE OF THE FACTORS OF X
WILL SIMPLIFY OUT. THIS WILL SIMPLIFY TO ONE, THIS
WILL SIMPLIFY TO X TO THE 1st. SO THIS FIRST PRODUCT
IS JUST 3X. SO F PRIME OF X=3X. AND HERE WE’LL JUST HAVE + 6X
NATURAL LOG 4X. WE NEED TO BE CAREFUL HERE. WE CAN’T ADD THIS 3X AND THIS 6X BECAUSE THIS 6X IS ATTACHED
TO THE NATURAL LOG BY MULTIPLICATION. SO THIS WOULD BE OUR FIRST
DERIVATIVE. AND NOW TO FIND THE SECOND
DERIVATIVE WE’LL FIND THE DERIVATIVE
OF THE FIRST DERIVATIVE, WHICH WE’LL DO ON THE NEXT
SLIDE. SO AGAIN, TO FIND THE SECOND
DERIVATIVE WE’LL TAKE THE DERIVATIVE
OF THE FIRST DERIVATIVE. THE DERIVATIVE OF 3X WOULD BE 3
PLUS– BUT NOW TO FIND THE DERIVATIVE
OF 6X x NATURAL LOG 4X WE HAVE TO APPLY
THE PRODUCT RULE AGAIN. SO THIS TIME WE’LL LET FIRST
FUNCTION F=6X AND THE SECOND FUNCTION G
=NATURAL LOG 4X. SO APPLYING THE PRODUCT RULE
AGAIN WE’LL HAVE F x G PRIME + G x F PRIME
OR THE FIRST FUNCTION x THE DERIVATIVE
OF THE SECOND FUNCTION + THE SECOND FUNCTION x THE DERIVATIVE OF THE FIRST
FUNCTION. SO WE HAVE F DOUBLE PRIME OF X=
3 + 6X x THE DERIVATIVE OF NATURAL LOG
4X. AGAIN, THIS IS A COMPOSITE
FUNCTION SO WE’LL LET THE INTER FUNCTION
4X=U. SO THE DERIVATIVE OF NATURAL LOG
U WITH RESPECT TO X, AGAIN, IS GOING TO BE 1/U
x U PRIME OR 1/4X x THE DERIVATIVE OF 4X
OR U PRIME, WHICH IS 4 + NATURAL LOG 4X
x THE DERIVATIVE OF 6X, WHICH WOULD JUST BE 6. NOW, AGAIN, LET’S SIMPLIFY. SO WE HAVE 3 + 4/1. 4/4 SIMPLIFIES TO 1. 6X x 1/X, NOW THE X’s SIMPLIFY
SO WE’RE LEFT WITH JUST 6. AND THEN WE HAVE + 6 NATURAL LOG
4X. WELL, WE CAN COMBINE THREE
AND SIX BECAUSE THOSE ARE LIKE TERMS. SO WE HAVE F DOUBLE PRIME OF X
=9 + 6 NATURAL LOG 4X. AGAIN, IT MIGHT BE TEMPTING TO
TRY TO ADD THE NINE AND THE SIX, BUT WE CANNOT DO THIS BECAUSE THE SIX IS ATTACHED
TO THE NATURAL LOG BY MULTIPLICATION. SO NOW WE HAVE OUR SECOND
DERIVATIVE FUNCTION. SO TO FIND THE THIRD DERIVATIVE WE’LL FIND THE DERIVATIVE
OF THE SECOND DERIVATIVE, WHICH AGAIN WE’LL DO
ON THE NEXT SLIDE. SO NOW TO FIND THE THIRD
DERIVATIVE, OR TRIPLE PRIME OF X, WE’LL FIND THE DERIVATIVE
OF THE SECOND DERIVATIVE. WELL, THE DERIVATIVE OF NINE
IS EQUAL TO ZERO. THE DERIVATIVE OF 6 NATURAL LOG
4X IS GOING TO BE EQUAL TO 6 x THE DERIVATIVE OF NATURAL LOG
4X, SO AGAIN THIS IS A COMPOSITE
FUNCTION. THE INTER FUNCTION IS EQUAL TO
U. THE DERIVATIVE OF NATURAL LOG U
WITH RESPECT TO X IS EQUAL TO 1/U x U PRIME, WHICH WOULD BE 1/4X x THE
DERIVATIVE OF 4X, WHICH IS 4. SO AGAIN THIS FOUR
AND THIS FOUR SIMPLIFY TO ONE. SO OUR THIRD DERIVATIVE IS EQUAL
TO 6 x 1/X OR 6 DIVIDED BY X. AND THAT’S GOING TO DO IT
FOR THIS EXAMPLE. I HOPE YOU FOUND THIS HELPFUL.