- TO DETERMINE THE DERIVATIVE
OF THE GIVEN FUNCTION
WE MUST RECOGNIZE THAT THE 
FUNCTION IS A QUOTIENT
OF TWO DIFFERENTIABLE FUNCTIONS.
SO IF WE LET THE NUMERATOR EQUAL 
F AND THE DENOMINATOR EQUAL G
WE CAN APPLY THE QUOTIENT RULE 
AS STATED HERE BELOW.
SO THE DERIVATIVE OF THE 
QUOTIENT OF F DIVIDED BY G
IS EQUAL TO G x F PRIME - F 
x G PRIME DIVIDED BY G SQUARED.
SO NOW LET'S GO AHEAD AND APPLY 
THE QUOTIENT RULE.
I ALWAYS LIKE TO START 
WITH THE DENOMINATOR
BECAUSE THE DENOMINATOR 
IS JUST GOING TO BE
THE DENOMINATOR SQUARED.
SO WE'LL HAVE THE QUANTITY 
2X SQUARED + 1
RAISED TO THE 2nd POWER
AND OUR NUMERATOR IS GOING TO BE 
G x F PRIME - F x G PRIME
OR THE DENOMINATOR
x THE DERIVATIVE 
OF THE NUMERATOR
WHICH WILL JUST BE 4 E TO THE X
MINUS THE NUMERATOR 
WHICH IS 4 E TO THE X
x THE DERIVATIVE OF THE 
DENOMINATOR WHICH WOULD BE 4X.
SO NOW WE HAVE THE DERIVATIVE 
FUNCTION
IT'S A MATTER OF PERFORMING 
ALGEBRA
TO SIMPLIFY THIS 
AS MUCH AS POSSIBLE.
WE'LL LEAVE OUR DENOMINATOR 
IN FACTORED FORM.
NOW WE'LL DISTRIBUTE HERE
SO WE'LL HAVE 8 X SQUARED 
E TO THE X
+ 4 E TO THE X
AND HERE WE'RE GOING TO HAVE 
- 16 X E TO THE X.
SO NOT MUCH IS GOING TO SIMPLIFY 
HERE
BUT LET'S GO AHEAD AND FACTOR 
OUT THE GREATEST COMMON FACTOR
OF THE NUMERATOR.
SO THE GREATEST COMMON FACTOR 
OF THESE THREE TERMS
WOULD BE 4 E TO THE X.
SO IF WE FACTOR OUT 4 E TO THE X 
FROM 8X SQUARED E TO THE X
WE'LL HAVE 2X SQUARED.
IF WE FACTOR OUT 4 E TO THE X 
FROM 4 E TO THE X WE'LL HAVE 1.
IF WE FACTOR OUT 4 E TO THE X 
FROM 16X E TO THE X WE'LL 4X.
NOW WE COULD WRITE THE TERMS 
OF THIS TRINOMIAL
IN DESCENDING ORDER
AND SEE IF IT FACTORS
BUT I CAN TELL BY LOOKING 
IT'S NOT GOING TO FACTOR.
SO WE HAVE OUR DERIVATIVE 
FUNCTION.
SO AS YOU BECOME MORE 
COMFORTABLE
WITH THE QUOTIENT RULE 
AND YOUR DERIVATIVE FORMULAS
HOPEFULLY YOU REALIZE 
THE MOST CHALLENGING PART
OF THESE TYPES OF PROBLEM
IS OFTEN THE ALGEBRA 
RATHER THAN THE CALCULUS.
I HOPE THIS WAS HELPFUL.
