TO DETERMINE THE DERIVATIVE 
OF THE GIVEN FUNCTION
YOU MIGHT BE THINKING ABOUT 
THE QUOTIENT RULE
BECAUSE WE DO HAVE A QUOTIENT.
BUT SINCE WE HAVE A CONSTANT 
AND A NUMERATOR
WE COULD CHANGE THE FORM 
OF THIS FUNCTION
SO WE CAN APPLY THE GENERAL 
OR EXTENDED POWER RULE
WHICH INCLUDE THE CHAIN RULE 
LISTED HERE BELOW.
SO WE CAN ELIMINATE THIS --
SO WE CAN ELIMINATE 
THE DENOMINATOR
IF WE MOVE THIS UP 
INTO THE NUMERATOR
WHICH WOULD CHANGE THIS EXPONENT 
TO -2.
SO WE'D HAVE F OF X EQUALS 2
x (3X SQUARED + 7 
RAISED TO THE POWER OF -2).
SO NOW WE HAVE 2 x A 
COMPOSITE FUNCTION.
AND ONCE WE KNOW WE HAVE 
A COMPOSITE FUNCTION,
WE WANT TO LET 
THE INNER FUNCTION EQUAL U.
SO HERE U IS GOING TO BE EQUAL 
TO 3X SQUARED + 7.
NOW, LET'S GO AHEAD 
AND WRITE THAT OVER HERE.
U = 3X TO THE 2nd + 7.
SO NOW WE CAN REWRITE 
THE GIVEN FUNCTION.
IN TERMS OF U WE WOULD HAVE 2U 
TO THE POWER OF -2.
SO NOW TO DETERMINE F PRIME OF X
WE NEED TO DETERMINE 
THE DERIVATIVE OF 2U TO THE -2
WITH RESPECTS TO U
AND THEN MULTIPLY IT BY U PRIME.
SO USING OUR EXTENDED POWER RULE
WE'RE GOING TO HAVE 2 x -2 x U 
TO THE -2 - 1 x U PRIME.
SO THIS WOULD BE THE DERIVATIVE 
OF THE GIVEN FUNCTION,
BUT NOW WE NEED TO REWRITE THIS 
IN TERMS OF X RATHER THAN U.
SO WE KNOW U = 3X SQUARED + 7, 
BUT WE ALSO NEED U PRIME.
SO U PRIME WOULD BE 6X.
SO NOW I'LL PERFORM 
THESE SUBSTITUTIONS
INTO OUR DERIVATIVE FUNCTION.
SO F PRIME OF X IS GOING TO BE 
EQUAL TO HERE WE HAVE -4,
AND THEN WE'LL HAVE INSTEAD OF U 
TO THE -3,
WE'LL HAVE 3X SQUARED + 7 
TO THE -3.
AND INSTEAD OF U PRIME 
WE'LL HAVE 6X.
AND NOW THERE IS ONE MORE STEP 
TO SIMPLIFY THIS.
AGAIN, WE CAN THINK OF 
ALL OF THIS AS BEING OVER ONE.
SO IF WE MOVE THIS QUANTITY 
TO THE DENOMINATOR,
IT WILL CHANGE THE SIGN 
OF THIS EXPONENT.
SO OUR DERIVATIVE FUNCTION WILL 
HAVE -4 x 6X IN THE NUMERATOR.
SO -24X.
AND OUR DENOMINATOR WILL BE
(3X SQUARED + 7 TO THE +3 POWER 
OR CUBED).
SO HERE IS THE FINAL FORM 
OF OUR DERIVATIVE FUNCTION.
SO AS YOU CAN SEE 
AS LONG AS WE CAN IDENTIFY THIS
IS A COMPOSITE FUNCTION,
LET U = THE INNER FUNCTION,
REWRITE THE FUNCTION 
IN TERMS OF U,
IT'S A FAIRLY 
STRAIGHT FORWARD PROCESS
BY APPLYING THE EXTENDED 
POWER RULE
WHICH INCLUDES THE CHAIN RULE.
WHICH BASICALLY MEANS WE FOUND 
THE DERIVATIVE OF 2U TO THE -2
WITH RESPECTS TO U
AND THEN MULTIPLIED BY U PRIME,
AND THEN WE HAD OUR DERIVATIVE.
AND THEN FROM THERE 
IT WAS ALGEBRA.
I HOPE THIS WAS HELPFUL.
