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X
- WANT TO DETERMINE THE DERIVATIVE
OF THE FOLLOWING FUNCTION USING THE POWER RULE,
WHICH TELLS US THAT THE DERIVATIVE OF X TO POWER OF N
IS EQUAL TO N x X TO THE POWER OF N - 1.
SO THIS CRAZY FUNCTION IS A NICE REVIEW
OF SOME OF THE STRATEGIES
THAT WE'LL USE IN ORDER TO APPLY THE POWER RULE
TO FIND THE DERIVATIVE OF A FUNCTION.
SO WE'LL FIRST REWRITE THIS FUNCTION
SO THAT WE CAN APPLY THE POWER RULE TO EACH OF TERM.
SO WE HAVE F OF X IS EQUAL TO--
WE MIGHT WANT TO REWRITE THIS FIRST TERM AS 1/5X TO THE 3 1/2.
AND THEN FOR THE SECOND TERM
WE NEED TO REWRITE THE CUBIT OF X IN RATIONAL EXPONENT FORM.
REMEMBER THIS X HAS AN EXPONENT OF 1,
SO THIS WOULD BE 4/X TO THE 1/3 POWER.
BUT THEN TO APPLY THE POWER RULE,
WE WANT THIS TO BE UP IN THE NUMERATOR.
SO IF WE MOVE THIS UP TO THE NUMERATOR
WE'RE GOING TO HAVE 4X TO THE POWER OF -1/3.
SO THAT'LL BE OUR SECOND TERM.
+ 4 X TO THE -1/3 POWER.
THEN WE CAN LEAVE THE LAST TWO TERMS THE WAY THEY ARE.
SO WE HAVE 7X SQUARED AND THEN - 9.
AND NOW WE CAN FIND THE DERIVATIVE OF EACH TERM.
THE DERIVATIVE OF 1/5X TO THE 3/2
IS GOING TO BE 1/5 x DERIVATIVE OF X TO THE 3/2,
SO WE'LL MULTIPLY BY THE EXPONENT.
AND THE NEW EXPONENT WILL BE 3/2 - 1 + THE DERIVATIVE OF 4X
TO THE -1/3 WILL BE 4 x THE DERIVATIVE OF X TO THE -1/3.
SO, AGAIN, WE'RE GOING TO MULTIPLY BY THE EXPONENT,
AND THEN WE'RE GOING TO SUBTRACT 1 TO GET THE NEW EXPONENT.
SO WE'LL HAVE -1/3 - 1
PLUS NEXT TERM WILL BE 7 x THE DERIVATIVE OF X TO THE SECOND,
WHICH IS 2 x X TO THE FIRST POWER MINUS THE DERIVATIVE OF 9.
BUT THE DERIVATIVE OF ANY CONSTANT IS ZERO.
SO WE'LL HAVE MINUS ZERO.
NOW THAT WE HAVE THE DERIVATIVE, THE REST IS ALGEBRA.
WE NEED TO SIMPLIFY THESE TERMS.
SO THE FIRST TERM, MULTIPLY THE FACTIONS,
WE'LL HAVE 3/10, X TO THE 3/2 - 1 IS 1/2.
HERE WE'D HAVE + 4 x -1/3, SO WE'LL WRITE THIS AS - 4/3,
X TO THE -4/3 + THIS WILL BE 14X.
AND THE LAST TERM IS 0.
NOW LET'S GO AHEAD AND CONVERT X TO THE 1/2 BACK TO A RADICAL,
AND THE SAME HERE.
SO F PRIME OF X IS GOING TO BE EQUAL TO 3 SQUARE ROOT X/10.
NOW, FOR HERE WE'RE GOING TO GO AHEAD
AND MOVE THIS DOWN TO THE DENOMINATOR
AND THEN WRITE IT AS A RADICAL.
SO WE'LL HAVE - 4, BUT 3 IS STILL A DENOMINATOR.
THIS WILL BE X TO THE +4/3 POWER AND THE DENOMINATOR.
SO THE INDEX WOULD BE 3 AND WE'D HAVE X TO THE FOURTH + 14X.
NOW WE'RE ALMOST THERE.
NOTICE HOW WE'RE TAKEN THE CUBE ROOT OF X TO THE FOURTH,
SO THERE'S A PERFECT CUBE FACTOR UNDERNEATH HERE OF X CUBED.
SO WE COULD SIMPLIFY THIS CUBE ROOT.
LET'S GO AHEAD AND DO THAT BEFORE WE STOP.
SO F PRIME OF X IS EQUAL TO 3 SQUARE ROOT X DIVIDED BY 10 - 4,
AND THE 3 IS HERE.
THEN BECAUSE THERE'S THREE FACTORS OF X UNDERNEATH HERE
WE'D HAVE X X THE CUBE ROOT OF X + 14X.
SO THIS WOULD BE OUR FINAL DERIVATIVE.
AND JUST IN CASE I LOST YOU ON SIMPLIFYING THIS CUBE ROOT,
THE CUBE ROOT OF X TO THE FOURTH
CONTAINS A PERFECT CUBE FACTOR RIGHT HERE,
WHICH SIMPLIFIES TO X CUBE ROOT X.
I HOPE YOU FOUND THIS EXAMPLE HELPFUL.