- 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.
