- NOW WE'LL LOOK AT SOME 
ADDITIONAL DERIVATIVES INVOLVING
THE NATURAL LOG FUNCTION 
OR LOG BASE E.
NORMALLY YOU'LL BE GIVEN 
TWO DIFFERENT FORMS
OF THE DERIVATIVE FORMULA FOR 
THE NATURAL LOG FUNCTION
GIVEN HERE.
NOTICE THE FIRST DERIVATIVE 
FORMULA
IS JUST FOR NATURAL LOG X.
THE DERIVATIVE OF NATURAL LOG X 
IS EQUAL TO 1 DIVIDED BY X
OR 1/X.
AND THEN WE HAVE THE DERIVATIVE 
OF NATURAL LOG U
WITH RESPECTS TO X 
= 1/U x U PRIME.
SO THIS FORMULA HERE INCLUDES 
THE CHAIN RULE
WHERE U WOULD BE 
A FUNCTION OF X.
AND THIS CAN ALSO BE WRITTEN 
AS U PRIME DIVIDED BY U.
BUT I LIKE THIS FORM 
BECAUSE IT REINFORCES THE IDEA
THAT WE'RE APPLYING 
THE CHAIN RULE.
NOW, IF YOU'RE EVER IN DOUBT
WHETHER YOU NEED TO APPLY 
THE CHAIN RULE,
YOU WOULD NEVER BE INCORRECT 
TO USE IT.
NOTICE IF WE TRIED TO USE THE 
CHAIN RULE FOR NATURAL LOG X,
U WOULD BE EQUAL TO X AND U 
PRIME WOULD JUST BE EQUAL TO 1.
LET'S TAKE A LOOK 
AT OUR FIRST EXAMPLE.
WE HAVE F OF X 
= 2 x NATURAL LOG X.
SO F PRIME OF X IS GOING TO BE 
EQUAL TO 2
x THE DERIVATIVE 
OF NATURAL LOG X,
WHICH IS 1/X OR 1 DIVIDED BY X.
YOU COULD THINK OF THIS 2 
AS BEING 2/1.
SO OUR DERIVATIVE FUNCTIONS 
IS 2 DIVIDED BY X.
OUR SECOND FUNCTION IS F OF X 
= NATURAL LOG 4X.
SO NOTICE HOW WE DO HAVE 
A COMPOSITE FUNCTION
WHERE THE INNER FUNCTION WOULD 
BE 4X.
SO WE'LL LET U = 4X.
SO ONCE WE IDENTIFY U WE'LL NEED 
TO DETERMINE DUDX OR U PRIME.
SO HERE U PRIME IS EQUAL TO 4.
SO NOW WE CAN REWRITE THIS 
FUNCTION AS NATURAL LOG U
AND THEN APPLY 
OUR DERIVATIVE FORMULA,
WHICH INCLUDES THE CHAIN RULE.
SO F PRIME OF X IS GOING TO BE 
EQUAL TO THE DERIVATIVE
OF NATURAL LOG U 
WITH RESPECTS TO X,
WHICH IS 1/U x U PRIME, 
WHICH WOULD BE 1/4X x 4 OR 4/1.
NOTICE HOW THESE FOURS SIMPLIFY 
OUT AND WE HAVE F PRIME OF X
IS EQUAL TO 1 DIVIDED BY X.
NOW, YOU MIGHT BE WONDERING 
HOW COULD THE DERIVATIVE
OF NATURAL LOG 4X BE THE SAME
AS THE DERIVATIVE OF NATURAL LOG 
X.
SO I DO WANT TO SHOW HOW YOU CAN 
DETERMINE THIS DERIVATIVE
USING A SECOND METHOD.
IF WE START THE ORIGINAL 
FUNCTION F OF X
= NATURAL LOG 4X,
WE COULD WRITE 4X AS 4 x X,
WHICH SHOULD REMIND US THAT
WE COULD APPLY A PROPERTY 
OF LOGARITHMS
TO REWRITE THIS LOG 
AS A SUM OF TWO LOGS.
AND THAT PROPERTY IS GIVEN HERE 
BELOW.
SO WE COULD WRITE F OF X = 
NATURAL LOG 4 + NATURAL LOG X.
AND, NOW, IF WE TAKE 
THE DERIVATIVE OF THE FUNCTION
IN THIS FORM, 
WE WOULD HAVE F PRIME OF X
IS EQUAL TO THE DERIVATIVE 
OF NATURAL LOG 4.
WELL, NATURAL LOG R 
IS A CONSTANT,
SO THIS DERIVATIVE IS EQUAL TO 0
+ THE DERIVATIVE OF NATURAL LOG 
X, WHICH IS JUST 1 DIVIDED BY X.
SO NOTICE HOW THE RESULT 
IS THE SAME DERIVATIVE FUNCTION,
FIRST, WHERE WE APPLIED 
THE CHAIN RULE,
AND SECOND WHERE WE APPLIED 
A PROPERTY OF LOGARITHMS
AND THEN FOUND THE DERIVATIVE.
AND NOTICE IN THIS FORM
WE DID NOT HAVE TO APPLY 
THE CHAIN RULE.
SO EITHER METHOD IS CORRECT,
BUT IT'S PROBABLY MORE COMMON 
TO USE THIS METHOD HERE
WHERE WE APPLY THE CHAIN RULE 
AND THEN SIMPLIFY THE RESULT.
WE'LL TAKE A LOOK AT MORE 
EXAMPLES IN THE NEXT VIDEO.
