- WE WANT TO DETERMINE THE DERIVATIVE
OF THE FOLLOWING FUNCTIONS USING THE POWER RULE.
THE POWER RULE STATES
THE DERIVATIVE OF X TO THE POWER OF N
IS EQUAL TO N x X TO THE POWER OF N - 1,
SO RIGHT AWAY WE SHOULD RECOGNIZE
THAT WE'RE NOT GOING TO BE ABLE TO APPLY THE POWER RULE
WITH THE FUNCTION IN THIS FORM.
WE NEED TO REWRITE THESE RADICALS
IN RATIONAL EXPONENT FORM BY USING THE RULE STATED HERE.
SO AS A QUICK EXAMPLE,
IF WE HAD SOMETHING LIKE THE CUBE ROOT OF X SQUARED
IN RATIONAL EXPONENT FORM,
THIS WOULD BE X TO THE POWER OF 2/3.
SO THE INDEX TELLS US THE DENOMINATOR
OF THE RATIONAL EXPONENT,
AND THE EXPONENT TELLS US THE NUMERATOR.
SO THAT'LL BE THE FIRST STEP IN DETERMINING
THE DERIVATIVES OF THESE TWO FUNCTIONS.
SO OUR FIRST FUNCTION IS F OF X = 9 x THE SQUARE ROOT OF X.
FOR A SQUARE ROOT, THE INDEX IS 2,
AND THEN THIS X HAS AN EXPONENT OF 1.
SO WE CAN REWRITE THE GIVEN FUNCTION AS F OF X = 9X
TO THE POWER OF 1/2.
NOW THAT WE HAVE X RAISED TO A POWER
WE CAN APPLY THE POWER RULE OF DIFFERENTIATION.
SO F PRIME OF X WILL EQUAL 9 x THE DERIVATIVE
OF X TO THE 1/2.
SO THE DERIVATIVE OF X TO THE 1/2
WE'RE GOING TO MULTIPLE BY 1/2,
THEN FOR THE NEW EXPONENT WE'LL HAVE 1/2 - 1.
SO 9 x 1/2 WOULD BE 9/2,
AND THEN WE'LL HAVE X TO THE POWER OF 1/2 - 1.
THAT WOULD BE X TO THE -1/2.
AND WE DON'T WANT TO LEAVE OUR DERIVATIVE IN THIS FORM
BECAUSE OF THE NEGATIVE EXPONENT,
SO IF WE MOVE THIS DOWN TO THE DENOMINATOR
IT WOULD BE X TO THE 1/2, SO LET'S GO AHEAD AND DO THAT.
WE'LL HAVE F PRIME OF X IS EQUAL TO 9
ALL OVER 2X TO THE POWER OF 1/2.
AND THE LAST THING WE WANT TO DO
IS REWRITE THIS BACK IN RADICAL FORM
SINCE THE ORIGINAL FUNCTION WAS IN RADICAL FORM.
AND WE ALREADY SAID BEFORE THAT THE SQUARE ROOT OF X
IS EQUAL TO X TO THE POWER OF 1/2,
SO IN RADICAL FORM THIS WOULD BE 9 ALL OVER 2,
SQUARE ROOT OF X.
NOTICE HOW THIS 1/2 IS ONLY ATTACHED TO THE X,
SO WE DON'T PUT THE 2 UNDERNEATH THE SQUARE ROOT.
OUR SECOND EXAMPLE IS GOING TO BE VERY SIMILAR.
WE'RE FIRST GOING TO WRITE THIS RADICAL
IN RATIONAL EXPONENT FORM,
SO F OF X IS EQUAL TO 15 x X TO THE POWER OF 2/5.
AGAIN, THE INDEX IS THE DENOMINATOR,
AND THE EXPONENT IS THE NUMERATOR.
AND NOW WE CAN APPLY THE POWER RULE,
F PRIME OF X IS GOING TO BE EQUAL TO 15 x THE DERIVATIVE
OF X TO THE 2/5,
SO WE'LL HAVE 2/5 x X TO THE 2/5 - 1.
NOW WE'LL SIMPLIFY.
HERE WHEN WE MULTIPLY THERE'S A COMMON FACTOR OF 5 HERE.
THERE'S 1, 5, AND 5 AND 3, 5'S AND 15.
SO THIS SIMPLIFIES TO 6X TO THE POWER OF 2/5 - 1 WOULD BE -3/5.
WE'LL FIRST SIMPLIFY BY MOVING X TO THE -3/5 DOWN
TO THE DENOMINATOR WHICH WILL CHANGE THE SIGN OF THE EXPONENT.
SO WE HAVE F PRIME OF X = 6 STAYS IN THE NUMERATOR.
WE HAVE X TO THE 3/5 IN THE DENOMINATOR.
AND THE LAST STEP, WE'LL WRITE THIS IN RADICAL FORM
BECAUSE THAT'S THE FORM WE STARTED WITH.
SO WE'LL HAVE 6 ALL OVER, THIS WILL BE THE 5TH ROOT,
THAT'S OUR INDEX,
AND WE HAVE X TO THE POWER OF 3 OR X TO THE 3rd.
SO OUR DENOMINATOR IS THE INDEX,
AND OUR NUMERATOR IS THE EXPONENT.
THERE WE GO.
IN THE NEXT VIDEO WE'LL LOOK AT ONE FUNCTION
AND USE A VARIETY OF THE TECHNIQUES
WE'VE DISCUSSED IN ORDER TO USE THE POWER RULE
TO FIND THE DERIVATIVE OF A FUNCTION.
HOPE YOU FOUND THIS HELPFUL.