- WE WANT TO DETERMINE THE DERIVATIVE
OF F OF X = 2X x THE QUANTITY 7X SQUARED - 2
RAISED TO THE 4th POWER.
WE CAN'T DISTRIBUTE THIS 2X,
BECAUSE THIS IS RAISED TO THE 4th POWER.
SO WE NEED TO RECOGNIZE THAT WE HAVE A PRODUCT OF TWO FUNCTIONS,
SO WE'LL HAVE TO APPLY THE PRODUCT RULE.
BUT ALSO IN DIFFERENTIATING THE SECOND FUNCTION,
WE'LL HAVE TO APPLY THE CHAIN RULE.
SO TO APPLY THE PRODUCT RULE,
LET'S LET THE FIRST FUNCTION EQUAL 2X.
WE'LL CALL IT F.
AND WE'LL LET THE SECOND FUNCTION
EQUAL THIS QUANTITY TO THE 4th, AND WE'LL CALL THIS G.
SO BEFORE WE ACTUALLY FIND ANY DERIVATIVES,
LET'S FIND OUT THE PRODUCT RULE.
SO F PRIME OF X IS GOING
TO BE EQUAL TO THE FIRST FUNCTION
WHICH IS 2X x THE DERIVATIVE OF THE SECOND FUNCTION
WHICH IS THE QUANTITY 7X SQUARED - 2
RAISED TO THE 4th + THE SECOND FUNCTION 7X SQUARED - 2
TO THE 4th x THE DERIVATIVE OF THE FIRST FUNCTION.
SO NOW, LET'S FOCUS ON DETERMINING
THIS FIRST DERIVATIVE HERE.
THIS IS A COMPOSITE FUNCTION,
SO WE'LL LET U EQUAL THE INNER FUNCTION.
SO WE'LL LET 7X SQUARED - 2 = U
WHICH MEANS THIS ENTIRE FUNCTION BECOMES U TO THE 4th.
SO TO DIFFERENTIATE U TO THE 4th,
WE'LL DIFFERENTIATE THIS WITH RESPECT TO U
AND THEN MULTIPLY IT BY U PRIME
APPLYING THE EXTENDED POWER RULE GIVEN HERE BELOW,
AND THEN OF COURSE THE DERIVATIVE OF 2X IS JUST 2.
SO LET'S WRITE OUT THE NEXT STEP.
WE'RE GOING TO HAVE 2X x THE DERIVATIVE OF U TO THE 4th,
THAT'LL BE 4U TO THE 3rd.
BUT INSTEAD OF WRITING U TO THE 3rd,
REMEMBER U = 7X SQUARED - 2, x U PRIME,
AND U PRIME IS A DERIVATIVE OF 7X SQUARED - 2
WHICH WOULD BE 14X + 7X SQUARED - 2 TO THE 4th x 2.
NOW THAT WE HAVE OUR DERIVATIVE, WE WANT TO TRY TO SIMPLIFY THIS.
SO IN THIS FIRST PART, IF WE CAN MULTIPLY 2X, 4 AND 14X,
THAT'S GOING TO BE 112X SQUARED x THE QUANTITY 7X SQUARED - 2
TO THE 3rd + THE SECOND PRODUCT IS JUST 2 x THE QUANTITY
7X SQUARED - 2 TO THE 4th POWER.
NOW TO FACTOR THE SUM,
WE NEED TO IDENTIFY THE GREATEST COMMON FACTOR.
AND THE GREATEST COMMON FACTOR OF THESE 2 PRODUCTS WOULD BE 2
AND THEN 3 FACTORS OF 7X SQUARED - 2.
SO WE WANT TO FACTOR THIS OUT OF BOTH OF THESE PRODUCTS.
SO LOOKING AT THIS FIRST PRODUCT,
IF WE FACTOR A 2 OUT 112X SQUARED, WE'D HAVE 56X SQUARED.
AND NOTICE THAT WE'RE FACTORING
ALL 3 FACTORS OF 7X SQUARED - 2 OUT,
SO WE'RE JUST LEFT WITH 56X SQUARED.
AND THEN ON THE SECOND PRODUCT,
WE'RE FACTORING OUT THE 2 AND ONLY 3
FACTORS OF 7X SQUARED - 2,
SO WE'RE LEFT WITH 1 FACTOR OF 7X SQUARED - 2.
AND THE LAST STEP IS TO COMBINE LIKE TERMS.
NOTICE HOW HERE WE HAVE 2X SQUARED TERMS.
SO OUR FINAL DERIVATIVE IS GOING TO BE
2 x THE QUANTITY 7X SQUARED - 2 TO THE 3rd x,
HERE WE'LL HAVE 63X SQUARED - 2,
AND THIS WOULD BE THE FACTORED FORM OF OUR DERIVATIVE.
SO LOOKING BACK AT WHAT WE DID,
NOTICE HOW ONCE WE APPLIED THE PRODUCT RULE AND THE CHAIN RULE
TO DETERMINE THIS DERIVATIVE HERE, FROM THIS STEP ON,
IT WAS ALL ALGEBRA.
WE'LL TAKE A LOOK AT TWO MORE EXAMPLES
THAT REQUIRE BOTH THE PRODUCT RULE AND THE CHAIN RULE
IN THE NEXT TWO VIDEOS.
I HOPE THIS EXAMPLE WAS HELPFUL.