- GIVEN F OF X = THE QUANTITY 3X 
SQUARED - 6X + 5
RAISED TO THE 3rd,
WE WANT TO FIND F PRIME OF X, 
THE DERIVATIVE FUNCTION,
AND F PRIME OF 2.
SO THE MOST IMPORTANT THING 
TO RECOGNIZE HERE IS THAT F OF X
IS A COMPOSITE FUNCTION,
AND THEREFORE, 
TO FIND THE DERIVATIVE,
WE HAVE TO APPLY THE CHAIN RULE.
SO BEFORE WE FIND 
THIS DERIVATIVE,
LET'S REVIEW THE CHAIN RULE.
THE CHAIN RULE IS USED 
TO FIND THE DERIVATIVE
OF A COMPOSITE FUNCTION,
WHERE IF WE HAVE Y 
= F OF G OF X WITH U
= TO THE INNER FUNCTION G OF X,
THEN THE CHAIN RULE CAN BE 
EXPRESSED SEVERAL WAYS.
HERE ARE THREE 
OF THE MOST COMMON WAYS.
TO FIND THE DERIVATIVE 
OF A COMPOSITE FUNCTION
WE NEED TO FIND THE DERIVATIVE 
OF THE OUTER FUNCTION,
AND THEN MULTIPLY 
BY THE DERIVATIVE
OF THE INNER FUNCTION.
SO LOOKING AT THESE NOTATIONS, 
DYDU, F PRIME OF G OF X,
AND F PRIME OF U 
REPRESENT THE DERIVATIVE
OF THE OUTER FUNCTION,
AND DUDX, G PRIME OF X, 
AND U PRIME
ALL REPRESENT THE DERIVATIVE 
OF THE INNER FUNCTION.
SO IF WE APPLY THE CHAIN RULE 
TO THE BASIC POWER RULE,
WE GET WHAT'S OFTEN CALLED 
THE CENTER POWER RULE,
OR GENERAL POWER RULE,
WHERE AGAIN, IF WE WANT 
TO FIND THE DERIVATIVE OF U
TO THE N WITH RESPECTS TO X,
WHERE U IS THE INNER FUNCTION,
IT'S EQUAL TO N x U TO 
THE POWER OF N - 1 x U PRIME.
WHERE AGAIN, THIS WOULD BE 
THE DERIVATIVE
OF THE OUTER FUNCTION,
AND THIS WOULD BE THE DERIVATIVE 
OF THE INNER FUNCTION.
SO GOING BACK TO OUR EXAMPLE,
THE FIRST STEP IS TO IDENTIFY 
THE INNER FUNCTION,
WHICH IN THIS CASE WOULD BE 
THE QUADRATIC FUNCTION
3X SQUARED - 6X + 5.
SO WE'RE GOING TO LET 
U = 3X SQUARED - 6X + 5.
LET'S ALSO FIND U PRIME NOW.
U PRIME, OR DUDX, 
WOULD BE 6X - 6.
LET'S GO AHEAD AND WRITE 
OUR FUNCTION IN TERMS OF U.
F OF X = U TO THE 3rd.
SO NOW WE'LL APPLY 
THE EXTENDED POWER RULE.
F PRIME OF X = 3 x U SQUARED 
x U PRIME.
WELL, WE ALREADY HAVE 
U AND U PRIME,
SO NOW WE'LL PERFORM 
THESE SUBSTITUTIONS.
F PRIME OF X = 3 
x THE QUANTITY 3X SQUARED - 6X
+ 5 SQUARED x U PRIME,
WHICH IS 6X - 6.
AGAIN, AFTER A WHILE 
YOU'LL PROBABLY GO
FROM -- FUNCTION 
TO THIS DERIVATIVE FUNCTION.
BUT I THINK WHEN FIRST 
LEARNING THE CHAIN RULE,
WRITING IT OUT IN TERMS OF U 
CAN BE VERY HELPFUL,
WHERE THESE FACTORS REPRESENT 
THE DERIVATIVE
OF THE OUTER FUNCTION,
AND THIS REPRESENTS 
THE DERIVATIVE
OF THE INNER FUNCTION.
BUT WE CAN SIMPLIFY THIS 
FURTHER.
WE CAN FACTOR OUT A 6 
FROM 6X - 6.
SO WE'D HAVE F PRIME OF X 
= 3 x 6,
LEAVING US WITH A FACTOR 
OF X - 1.
AND WE STILL HAVE 
3X SQUARED - 6X + 5 SQUARED.
SO WE HAVE F PRIME OF X 
= 18 x THE QUANTITY X - 1
x THE QUANTITY 3X SQUARED - 6X 
+ 5 SQUARED.
SO THIS IS THE FIRST PART 
OF OUR PROBLEM,
THE DERIVATIVE FUNCTION.
SO NOW TO FIND F PRIME OF 2, 
WE'LL SUBSTITUTE 2 FOR X.
F PRIME OF 2 WOULD GIVE US 
THE SLOPE OF THE TANGENT LINE
ON X = 2.
IT WILL ALSO TELL US THE 
INSTANTANEOUS RATE OF CHANGE
OF THE FUNCTION AT X = 2.
SO WE'D HAVE 18 x 2 - 1 x 3 x 
2 SQUARED - 6 x 2 + 5 SQUARED.
SO WE'LL HAVE 18 x 1 
x 12 - 12 = 0 + 5 SQUARED,
AND THAT'D BE 18 x 25 OR 450.
SO F PRIME OF 2 = 450.
LET'S CHECK THE TANGENT LINE 
AT X = 2
TO SEE IF IT LOOKS 
LIKE IT HAS A SLOPE OF 450.
AGAIN, WE FOUND 
THAT F PRIME OF 2 = 450.
NOTICE WHEN X = 2 THE POINT 
ON THE FUNCTION WOULD BE 2,125.
WE'D FIND THE Y COORDINATE 
BY SUBBING 2
INTO THE ORIGINAL FUNCTION HERE.
AND NOTICE THE SLOPE OF 
THE TANGENT LINE AT THAT POINT
IS VERY STEEP,
GOING UPWARD FROM LEFT TO RIGHT.
AND BECAUSE F PRIME OF 2 = 450, 
THE SLOPE OF THIS LINE IS 450.
I HOPE YOU FOUND THIS HELPFUL.
