- WE'RE GIVEN A FUNCTION THAT CONTAINS RADICALS
AND ASKED TO FIND ITS DERIVATIVE OR F PRIME OF X.
TO FIND THE DERIVATIVE FUNCTION,
WE'LL BE APPLYING THE POWER RULE OF DIFFERENTIATION GIVEN HERE,
BUT BEFORE WE DO THIS WE'LL HAVE TO FIRST WRITE THE RADICALS
USING RATIONAL EXPONENTS
AND THEN APPLY THESE PROPERTIES OF EXPONENTS
SO THAT WE CAN HAVE EACH TERM IN THE FORM
OF X RAISED TO SOME POWER OF N
AND THEN APPLY OUR DERIVATIVE RULE.
SO TO BEGIN,
NOTICE HOW WE CAN WRITE A RADICAL USING RATIONAL EXPONENTS
WHERE THE EXPONENT WOULD BE THE NUMERATOR OF THE EXPONENT
AND THE INDEX WOULD BE THE DENOMINATOR OF THE EXPONENT.
SO LOOKING AT OUR FUNCTION
WHEN WE HAVE THE SQUARE ROOT OF X THE INDEX IS 2
AND THE EXPONENT IS 1.
WHEN WE HAVE THE CUBE ROOT OF X
WE CAN SEE THE INDEX IS 3 AND THE EXPONENT IS 1.
SO WE CAN REWRITE THE GIVEN FUNCTION
AS F OF X = 2 X TO THE 4th TIMES--
THIS WOULD BE X TO THE 1/2
AND THEN -4 DIVIDED BY X TO THE 2nd x X TO THE 1/3.
NOW WE'RE GOING TO FIND THIS PRODUCT AND THIS PRODUCT,
AND NOTICE HOW THE BASES ARE THE SAME.
SO WE'LL BE APPLYING THIS RULE HERE WHERE WE ADD THE EXPONENTS.
SO WE'D HAVE F OF X = 2 x X TO THE POWER OF 4 + 1/2.
WELL, OF COURSE 4 + 1/2 WOULD BE 4 1/2,
BUT AS AN IMPROPER FRACTION
THIS WOULD BE 2 x 4 + 1, THAT'S 9, 9/2.
SO WE HAVE 2X TO THE 9/2 AND THEN HERE WE'LL HAVE
-4 DIVIDED BY X TO THE POWER OF 2 + 1/3.
WELL, 2 + 1/3, OF COURSE, IS 2 1/3.
AS AN IMPROPER FRACTION THOUGH,
WE'D HAVE 3 x 2, THAT'S 6 + 1 = 7, 7/3.
SO WE HAVE 4 DIVIDED BY X TO THE 7/3.
NOW THERE'S ONE MORE THING WE STILL HAVE TO DO, THOUGH.
WE STILL HAVE TO CHANGE THE FORM OF THIS TERM HERE
TO APPLY THE POWER RULE.
WE'RE GOING TO MOVE X TO THE 7/3 UP TO THE NUMERATOR,
AND BY DOING THIS IT'S GOING TO CHANGE
THE SIGN OF THE EXPONENT
WHICH IS A VARIATION OF THIS EXPONENT RULE HERE.
SO FINALLY WE HAVE
F OF X = 2X TO THE 9/2 - 4X TO THE POWER OF -7/3.
AND NOW WE CAN FIND THE DERIVATIVE FUNCTION
BY APPLYING THE POWER RULE OF DIFFERENTIATION.
F PRIME OF X IS GOING TO = 2 x THE DERIVATIVE OF X TO THE 9/2.
THIS IS GOING TO BE 2 x 9/2X TO THE POWER OF 9/2
- 1 - 4 x THE DERIVATIVE OF X TO THE POWER OF -7/3.
THAT'S GOING TO BE x -7/3, X TO THE POWER OF -7/3 - 1.
SO NOW WE'LL GO AHEAD AND SIMPLIFY.
NOTICE HERE THIS IS 2/1, SO THESE 2'S SIMPLIFY OUT.
SO WE HAVE F PRIME OF X = 9 x X TO THE POWER OF 9/2 - 1.
WELL, 9/2 - 1/1, OF COURSE, IS THE SAME AS 9/2 - 2/2 OR 7/2.
SO WE HAVE 9X TO THE 7/2.
HERE WE HAVE A NEGATIVE TIMES A NEGATIVE.
THAT'S GOING TO BE POSITIVE SO PLUS,
THIS IS THE SAME AS 4/1, SO THAT'LL BE 28/3,
AND THEN WE HAVE X TO THE POWER OF -7/3 - 1,
-7/3 - 1 IS THE SAME AS -7/3 - 3/3 WHICH WOULD BE -10/3.
SO WE HAVE 28/3 x X TO THE -10/3.
SO THIS WOULD BE OUR DERIVATIVE FUNCTION,
BUT LET'S GO AHEAD AND REWRITE THIS SECOND TERM
USING A POSITIVE EXPONENT.
WE COULD SAY F PRIME OF X = 9 x X TO THE 7/2
AND THEN + WE'RE GOING TO MOVE THIS TO THE DENOMINATOR
WHICH WILL CHANGE THE SIGN OF THE EXPONENT,
SO WE'LL HAVE 28 DIVIDED BY 3X TO THE POWER OF 10/3.
NOW WE SHOULD BE FINE TO LEAVE IT IN THIS FORM HERE,
BUT I ALSO DO WANT TO SHOW
HOW WE COULD REWRITE THIS BACK IN RADICAL FORM.
SO LET'S GO AHEAD AND CONTINUE THIS ON THE NEXT SLIDE.
WRITING THIS BACK IN RADICAL FORM,
NOTICE HOW THE DENOMINATOR HERE IS 2,
SO THIS CAN BE WRITTEN AS A SQUARE ROOT.
THE DENOMINATOR HERE IS 3,
SO THIS CAN BE WRITTEN AS A CUBE ROOT.
SO WE COULD WRITE THIS AS
F PRIME OF X = 9 x THE SQUARE ROOT OF X TO THE 7th
+ 28 DIVIDED BY 3 x THE CUBE ROOT OF X TO THE 10th,
BUT THESE RADICALS CAN BE SIMPLIFIED.
X TO THE 7th = X TO THE 6th x X
WHERE X TO THE 6th IS A PERFECT SQUARE ROOT,
SO WE CAN SIMPLIFY THIS AS
F PRIME OF X = 9X CUBED x THE SQUARE ROOT OF X.
AGAIN, THIS IS BECAUSE WE CAN WRITE X TO THE 7th
AS X TO THE 6th x X
WHERE THE SQUARE ROOT OF X TO THE 6th = X TO THE 3rd.
AND THEN WE HAVE + 28 DIVIDED BY 3.
THEN THE CUBE ROOT OF X TO THE 10th
CAN SIMPLIFY BECAUSE X TO THE 9th IS A PERFECT CUBE.
IT WOULD BE X TO THE 3rd x THE CUBE ROOT OF X.
AGAIN, BECAUSE WE CAN WRITE THE CUBE ROOT OF X TO THE 10th
AS X TO THE 9th x X
AND THE CUBE ROOT OF X TO THE 9th = X TO THE 3rd.
SO AGAIN, IT'S PROBABLY NOT NECESSARY,
BUT JUST IN CASE YOU'RE REQUIRED
TO WRITE IT BACK IN RADICAL FORM, HERE IT IS.
I HOPE YOU FOUND THIS HELPFUL.