- TO DETERMINE THE DERIVATIVE 
OF THE GIVEN FUNCTION
WE MUST FIRST RECOGNIZE THAT WE 
HAVE A PRODUCT OF TWO FUNCTIONS
WE HAVE 6 TO THE POWER OF X 
x LOG BASE 3 OF X.
SO WE'LL HAVE TO APPLY THE 
PRODUCT RULE GIVEN HERE.
WE'LL ALSO HAVE TO USE THE 
DERIVATIVE FORMULA
FOR "A" TO THE X AND FOR LOG 
BASE "A" OF X GIVEN HERE.
LET'S START BY SETTING UP THE 
PRODUCT RULE.
IF THEY CALL THE FIRST FUNCTION 
F AND THE SECOND FUNCTION G
THE DERIVATIVE WILL BE EQUAL TO 
F x G PRIME + G x F PRIME
FOR THE FIRST FUNCTION
x THE DERIVATIVE OF THE SECOND 
FUNCTION + THE SECOND FUNCTION
x THE DERIVATIVE 
OF THE FIRST FUNCTION.
SO I ALWAYS THINK IT'S A GOOD 
IDEA TO SHOW THIS STEP HERE
BEFORE WE ACTUALLY DETERMINE 
ANY DERIVATIVES.
SO NOW WE'LL DETERMINE THE 
DERIVATIVE OF THE LOG FUNCTION
HERE AND THE DERIVATIVE OF THE 
EXPONENTIAL FUNCTION HERE.
SO WE'LL HAVE F PRIME OF X IS 
EQUAL TO 6 TO THE POWER OF X x--
FOR THE DERIVATIVE OF LOG BASE 3 
OF X
NOTICE HOW "A" IS EQUAL TO 3.
SO WE'LL HAVE 1/NATURAL LOG 3 
x 1/X
+ LOG BASE 3 OF X
x THE DERIVATIVE OF 6 
TO THE POWER OF X.
SINCE OUR BASE IS EQUAL TO 6 
NOTICE THAT "A" IS EQUAL TO 6
SO WE'LL HAVE NATURAL LOG 6 
x 6 TO THE POWER OF X.
NOW LET'S DETERMINE 
THESE PRODUCTS.
THIS FIRST PRODUCT WILL BE 
A FRACTION
WHERE THE NUMERATOR IS GOING TO 
BE 6 TO THE POWER OF X.
DENOMINATOR WOULD BE 
X NATURAL LOG 3 +--
TO KEEP THIS PRODUCT STRAIGHT
WE MAY WANT TO PUT EVERYTHING 
IN PARENTHESES.
SO WE HAVE (LOG BASE 3 OF X) x 
(6 TO THE X) x (NATURAL LOG 6).
SO WE COULD TRY TO GET A COMMON 
DENOMINATOR AND COMBINE THESE
BUT I THINK FOR THIS EXAMPLE 
WE'LL LEAVE IT IN THIS FORM.
THIS WOULD BE OUR 
DERIVATIVE FUNCTION.
