- WELCOME TO AN INTRODUCTION TO THE PRODUCT
RULE OF DIFFERENTIATION.
THE PRODUCT RULE GIVES US A FORMULA TO FIND THE DERIVATIVE
WHEN WE HAVE THE PRODUCT OF 2 FUNCTIONS SEEN HERE
USING FUNCTION NOTATION AND SEEN HERE JUST USING F AND G.
THE DERIVATIVE OF A PRODUCT OF 2 FUNCTIONS IS THE FIRST FUNCTION
WHICH IS F, x THE DERIVATIVE OF THE SECOND FUNCTION
WHICH IS G PRIME + THE SECOND FUNCTION WHICH IS G,
x THE DERIVATIVE OF THE FIRST FUNCTION WHICH IS F PRIME.
NOW I DID MAKE A VIDEO THAT PROVES THE PRODUCT RULE.
BUT LET'S START BY LOOKING AT A SIMPLE EXAMPLE.
WE KNOW THE DERIVATIVE OF X TO THE FIFTH
APPLYING THE POWER RULE WOULD BE 5X TO THE 1/4.
BUT IF WE WANTED TO, WE COULD WRITE X TO THE FIFTH
AS OUR PRODUCT OF TWO FUNCTIONS.
WE COULD WRITE THIS AS X TO THE SECOND x X TO THE THIRD.
SO IF WE APPLIED THAT PRODUCT RULE IN THIS FORM,
WE'D HAVE THE FIRST FUNCTION WHICH IS X SQUARED,
x THE DERIVATIVE OF THE SECOND FUNCTION,
WHICH IS X TO THE THIRD + THE SECOND FUNCTION
WHICH IS X TO THE THIRD, x THE DERIVATIVE OF THE FIRST FUNCTION
WHICH WOULD BE THE DERIVATIVE OF X TO THE SECOND.
SO LET'S GO AHEAD AND FIND THESE TWO DERIVATIVES
AND SIMPLIFY TO MAKE SURE WE STILL GET 5X TO THE FOURTH
WHICH WE KNOW TO BE THE DERIVATIVE.
SO HERE WE'D HAVE X SQUARED x THE DERIVATIVE OF X CUBED
WOULD BE 3X SQUARED + X TO THE THIRD
x THE DERIVATIVE OF X TO THE SECOND, WHICH WOULD JUST BE 2X.
WELL THIS WOULD BE 3X TO THE FOURTH, + 2X TO THE FOURTH,
WHICH DOES HAVE A SUM OF 5X TO THE FOURTH.
NOTICE HOW WHEN WE WROTE X TO THE FIFTH,
THE PRODUCT OF TWO FUNCTIONS, AND THEN FOUND THE DERIVATIVE,
IT'S THE SAME AS IF WE APPLIED THE POWER RULE TO X OF THE FIFTH
WHICH IS WHAT WE EXPECT.
NOW WE ACTUALLY WOULD NEVER DO THIS,
BUT THIS IS A NICE VERIFICATION THAT THE PRODUCT RULE DOES WORK.
LET'S TAKE A LOOK AT A MORE INVOLVED EXAMPLE.
WE WANT TO FIND THE DERIVATIVE OF F OF X = THE QUANTITY 4X - 3
x THE QUANTITY 2X SQUARED + 3X + 5.
SO THIS WILL BE OUR FIRST FUNCTION. WE'LL CALL IT F,
AND THIS WILL BE OUR SECOND FUNCTION, WE'LL CALL IT G.
AND FOR REFERENCE WE HAVE THE PRODUCT RULE DOWN HERE BELOW.
SO WE'RE GOING TO SET UP THE PRODUCT RULE
AND THEN WELL FIND THE DERIVATIVES IN THE SECOND STEP.
SO F PRIME OF X MAY BE EQUAL TO THE FIRST FUNCTION
WHICH IS THE QUANTITY 4X - 3
x THE DERIVATIVE OF THE SECOND FUNCTION
WHICH IS 2X SQUARED + 3X + 5.
AGAIN WE'LL FIND THE DERIVATIVE IN THE SECOND STEP
PLUS THE SECOND FUNCTION WHICH IS 2X SQUARED + 3X + 5
x THE DERIVATIVE OF THE FIRST FUNCTION, WHICH IS 4X - 3.
SO AGAIN HERE WE HAVE THE FIRST FUNCTION,
F x THE DERIVATIVE OF THE SECOND FUNCTION + THE SECOND FUNCTION
x THE DERIVATIVE OF THE FIRST FUNCTION.
SO HERE'S OUR PRODUCT RULE.
NOW WE'RE GOING TO FIND THIS DERIVATIVE
AND THEN THIS DERIVATIVE THEN MULTIPLY AND SIMPLIFY.
SO F PRIME OF X IS GOING TO BE EQUAL TO THE QUANTITY 4X - 3
x THIS DERIVATIVE WHICH WOULD BE 4X + 3
+ THE QUANTITY 2X SQUARED + 3X + 5 x THE DERIVATIVE OF 4X - 3
WHICH WOULD JUST BE 4.
NOW WE'LL FIND THE 2 PRODUCTS AND COMBINE LIKE TERMS.
SO REMEMBER HERE WE'LL HAVE 4 PRODUCTS, 1, 2, 3, 4
AND HERE WE'LL DISTRIBUTE THE 4, SO WE'LL HAVE 1, 2, 3 PRODUCTS.
SO HERE WE HAVE 16X SQUARED,
AND THESE ARE ACTUALLY CONJUGATES.
SO WE'RE GOING TO HAVE + 12X AND THEN - 12X,
THAT'S GOING TO BE 0.
AND THEN -3 x 3 IS - 9 + DISTRIBUTING 4,
WE'D HAVE 8X SQUARED + 12X AND THEN 20.
NOW IT'S -- LIKE TERMS WE HAVE 2X SQUARED TERMS
AND WE ALSO HAVE 2 CONSTANTS.
SO OUR DERIVATIVE WHERE F PRIME OF X IS EQUAL TO,
THAT WOULD BE 24X SQUARED,
+ 12X AND THEN -9 + 20 IS EQUAL TO 11, SO WE HAVE + 11.
THIS IS OUR DERIVATIVE FUNCTION. NOW YOU MIGHT BE WONDERING
IF WE COULD HAVE JUST MULTIPLIED ALL OF THIS OUT
AND THEN JUST APPLIED THE BASIC POWER RULE.
IN THIS CASE WE ACTUALLY CAN
SO I'M GOING TO GO AHEAD AND SHOW
WHAT WOULD HAPPEN IF WE DID MULTIPLY IT OUT
JUST TO SHOW WE GET THE SAME ANSWER.
AND I ALREADY WROTE THIS OUT.
IF WE MULTIPLY THIS OUT, THIS WOULD BE OUR PRODUCT,
COMBINING LIKE TERMS THIS WOULD BE OUR SIMPLIFIED FUNCTION
AND NOW IF WE APPLY THE BASIC POWER RULE,
NOTICE HOW WE WOULD JUST HAVE 24X SQUARED, + 12X + 11
AND THE DERIVATIVE OF -15 WOULD BE 0.
SO THIS IS THE SAME DERIVATIVE WE HAD ON THE PREVIOUS SLIDE.
BUT IN MANY CASES MULTIPLYING IT OUT WOULD BE A LOT MORE WORK
AND THAT'S WHY THE PRODUCT RULE IS SO HELPFUL.
LET'S TAKE A LOOK AT ONE MORE EXAMPLE.
AGAIN THIS IS ONE WHERE WE COULD MULTIPLY IT OUT
BUT THERE WOULD BE 9 DIFFERENT PRODUCTS TO FIND
AND THEN SIMPLIFY BEFORE WE COULD EVEN
THINK OF APPLYING THE POWER RULE.
SO WE'LL GO AHEAD AND APPLY THE PRODUCT RULE
SO WE'RE GOING TO LET THIS FIRST FUNCTION BE FUNCTION F
AND THE SECOND FUNCTION BE FUNCTION G
AND NOW WE'LL APPLY THE PRODUCT RULE.
SO OUR DERIVATIVE IS GOING TO BE EQUAL TO THE FIRST FUNCTION,
F x THE DERIVATIVE OF THE SECOND FUNCTION,
WHICH IS X SQUARED + 3X - 9,
+ THE SECOND FUNCTION x THE DERIVATIVE
OF THE FIRST FUNCTION.
AGAIN NOTICE HOW I'M NOT FINDING ANY DERIVATIVES
IN THE FIRST STEP.
I'M JUST SETTING UP THE PRODUCT RULE.
NOW WE'LL GO BACK AND FIND THE DERIVATIVE HERE,
THE DERIVATIVE HERE, MULTIPLY AND THEN SIMPLIFY.
THE DERIVATIVE OF X SQUARED + 3X - 9 IS GOING TO BE 2X + 3
AND THEN WE HAVE THE SECOND FUNCTION
x THE DERIVATIVE OF THE FIRST FUNCTION.
THIS SHOULD BE 6X - 1.
NOW WE'LL MULTIPLY THIS OUT AND COMBINE LIKE TERMS.
AND TO HELP US SIMPLIFY THIS,
I'M GOING TO LINE UP THE LIKE TERMS VERTICALLY.
WE HAVE 3X SQUARED x 2X THAT'S 6X CUBED.
3X SQUARED x 3 THAT'S + 9X SQUARED.
THEN WE HAVE -X x 2X THAT'S GOING TO BE -2X SQUARED
WHICH AGAIN I'M GOING TO PUT
UNDERNEATH THE X SQUARED TERM HERE.
-X x 3 THAT'S - 3X AND WE HAVE 5 x 2X THAT'S + 10X,
AGAIN I'LL PLACE THIS UNDER THE X TERM
AND WE HAVE 5 x 3 THAT'S + 15
AND NOW WE HAVE 6 MORE PRODUCTS HERE, X SQUARED x 6X,
THAT'S + 6X CUBED.
X SQUARED x - THAT'S GOING TO BE -1X SQUARED OR -X SQUARED.
AGAIN NOTICE HOW I'M PUTTING IT UNDER THE X SQUARED COLUMN.
THEN WE HAVE 3X x 6X THAT'S + 18 X SQUARED.
3X x -1 THAT'S -3X OR - 3X.
THEN WE HAVE -9 x 6X THAT'S -54X
OR - 54X AND THEN -9 x -1 THAT'S + 9.
OKAY, NOW WE'LL COMBINE OUR LIKE TERMS VERTICALLY
TO FIND OUR SIMPLIFIED DERIVATIVE FUNCTION.
SO WE HAVE 12X CUBED THEN WE HAVE 9X SQUARED - 2X SQUARED,
- 1X SQUARED, THAT'S 6X SQUARED, + 18 X SQUARED,
THAT'S + 24X SQUARED.
LOOKING AT THE X TERMS,
-3X + 10X - 3X THAT'S GOING TO BE 4X - 54X
THAT'S GOING TO BE - 50X AND THEN 15 + 9 IS 24.
SO YOU CAN SEE WE NEED TO BE VERY CAREFUL
WHEN MULTIPLYING THIS OUT,
WATCHING OUR SIGNS AND MAKE SURE WE COMBINE THE LIKE TERMS.
OKAY, I HOPE YOU FOUND THESE FIRST TWO EXAMPLES HELPFUL.
THANK YOU FOR WATCHING.