- IF F OF X = 
THE GIVEN THE FUNCTION,
WE WANT TO FIND THE DERIVATIVE 
FUNCTION OR F PRIME OF X
AND THEN FIND F PRIME OF 1.
NOTICE HOW THE FUNCTION IS GIVEN 
AS A PRODUCT OF 2 FUNCTIONS.
SO IF WE WANT TO FIND
THE DERIVATIVE FUNCTION IN THIS 
FORM, WE'LL HAVE TO APPLY
THE PRODUCT RULE OF 
DIFFERENTIATION GIVEN HERE BELOW
WHERE THE DERIVATIVE OF FUNCTION 
F x FUNCTION G
IS EQUAL TO F x G PRIME + G x F 
PRIME,
OR WE CAN SAY THE FIRST FUNCTION
x THE DERIVATIVE OF THE SECOND 
FUNCTION + THE SECOND FUNCTION
x THE DERIVATIVE OF THE FIRST 
FUNCTION.
OF COURSE, THE OTHER OPTION
WOULD BE TO FIND 
THIS PRODUCT FIRST
AND THEN FIND THE DERIVATIVE 
FUNCTION
USING THE BASIC POWER RULE.
THE MAIN PURPOSE IS TO INTRODUCE 
THE PRODUCT RULE,
SO WE'LL GO AHEAD 
AND APPLY THE PRODUCT RULE
TO FIND OUR DERIVATIVE.
BUT TOWARD THE END OF THE VIDEO,
WE'LL ALSO SHOW THAT 
IF WE MULTIPLY THIS OUT
AND THEN FIND THE DERIVATIVE, 
THE RESULT WOULD BE THE SAME.
SO USING OUR FORMULA GIVEN 
HERE BELOW,
WE'LL LET THE FIRST FUNCTION BE 
FUNCTION F
AND THE SECOND FUNCTION 
BE FUNCTION G.
AND NOW, WE'LL SET UP 
THE PRODUCT RULE.
SO F PRIME OF X IS GOING TO BE 
EQUAL TO F x G PRIME
OR THE FIRST FUNCTION
x THE DERIVATIVE OF THE SECOND 
FUNCTION + G x F PRIME
WHICH SHOULD BE THE SECOND 
FUNCTION
x THE DERIVATIVE OF THE FIRST 
FUNCTION.
SO NOW, WE'LL FIND THE 
DERIVATIVE HERE AND HERE,
THEN FIND THE PRODUCTS 
AND THEN SIMPLIFY.
SO F PRIME OF X IS GOING 
TO BE EQUAL TO,
AGAIN, THE FIRST FUNCTION x THE 
DERIVATIVE OF 3X SQUARED - 2.
SO TO FIND THE DERIVATIVE 
OF 3X SQUARED,
WE'RE GOING TO MULTIPLY BY 2, 
THAT WOULD GIVE US 6.
SUBTRACT 1 FROM THE EXPONENT,
THAT WOULD BE X TO THE 1st 
WHICH IS X - A DERIVATIVE OF 2
WHICH WOULD BE 0, 
SO OUR DERIVATIVE IS JUST 6X,
THEN WE'LL HAVE + 3X SQUARED -2 
x THE DERIVATIVE OF X SQUARED
- 5X + 8.
THE DERIVATIVE OF X SQUARED
WOULD BE 2X - THE DERIVATIVE 
OF 5X
WHICH WOULD JUST BE 5 AND THE 
DERIVATIVE OF 8 WOULD BE 0,
SO OUR DERIVATIVE IS 2X - 5.
SO NOW, WE'LL FIND 
THESE PRODUCTS.
HERE WE'LL HAVE 3 PRODUCTS, 
1, 2, 3,
AND HERE WE'LL HAVE 4 PRODUCTS, 
1, 2, 3, AND 4.
SO WE HAVE F PRIME OF X = 6X 
TO THE 3rd - 30X SQUARED + 48X.
NOW FOR THIS PRODUCT,
WE'LL HAVE +6X TO THE 3rd - 15X 
SQUARED - 4X
AND THEN FINALLY 10.
NOW, WE'LL GO AHEAD AND COMBINE 
LIKE TERMS.
THERE ARE 2X TO THE 3rd TERMS, 
THERE ARE 2X SQUARED TERMS,
AND THERE ARE ALSO 2X TERMS.
SO F PRIME OF X IS EQUAL TO 12X 
TO THE 3rd,
THEN WE'D HAVE -45X SQUARED, 
THEN WE'D HAVE + 44X + 10.
SO THIS IS OUR DERIVATIVE 
FUNCTION.
SO TO FIND F PRIME OF 1
WHICH WOULD BE THE SLOPE 
OF THE TANGENT LINE AT X = 1,
WE'LL SUBSTITUTE 1 FOR X 
INTO OUR DERIVATIVE FUNCTION.
SO WE HAVE 12 - 45, 
THAT'S -33 + 44,
THAT'S 11 + 10, SO WE HAVE PRIME 
OF 1 IS EQUAL TO 21
WHICH AGAIN WOULD BE THE SLOPE 
OF THE TANGENT LINE WHEN X = 1.
SO LET'S GO AHEAD 
AND TAKE A LOOK AT A GRAPH.
THE BLUE GRAPH IS THE GRAPH 
OF THE ORIGINAL FUNCTION.
WHEN X = 1, WE'D BE AT THIS 
POINT HERE ON THE FUNCTION.
THE RED LINE 
WOULD BE THE TANGENT LINE.
NOTICE HOW THE Y-AXIS 
IS SCALED BY 5s,
SO THE SLOPE OF THIS RED TANGENT 
LINE DOES APPEAR TO BE 21
WHICH VERIFIES THAT OUR WORK 
IS CORRECT.
NOW, BEFORE WE GO, 
I DO WANT TO GO BACK AND SHOW
HOW IF WE WERE TO FIND 
THE DERIVATIVE
BY FIRST FINDING THIS PRODUCT,
THE DERIVATIVE WOULD BE THE SAME
AS WE FOUND 
USING THE PRODUCT RULE.
I'VE ALREADY WORKED THIS OUT, 
BUT I DO WANT TO SHOW IT.
AGAIN IF WE WERE TO START
WITH THE ORIGINAL FUNCTION
AND MULTIPLY THIS OUT, WE WOULD 
HAVE A TOTAL OF 6 PRODUCTS,
1, 2, 3, 4, 5, AND 6,
WHICH WOULD GIVE US 
THESE TERMS HERE.
AND IF WE COMBINE LIKE TERMS, 
IT WOULD GIVE US THIS FUNCTION.
THIS IS THE SAME AS THE ORIGINAL 
FUNCTION,
BUT WE'VE EXPANDED THE PRODUCT.
AND NOW IF WE APPLY THAT PRODUCT 
RULE TO FIND
THE DERIVATIVE OF THE FUNCTION 
IN THIS FORM,
NOTICE HOW THE DERIVATIVE 
FUNCTION IS STILL THE SAME.
WE HAVE F PRIME OF X = 12X CUBED 
- 45X SQUARED + 44X + 10.
BUT AGAIN, THE MAIN IDEA 
FOR THIS VIDEO WAS TO SHOW
HOW TO USE THE PRODUCT RULE.
I HOPE YOU FOUND THIS HELPFUL.
