- SOMETIMES THERE'S CONFUSION 
ABOUT THE DIFFERENCE
BETWEEN THESE TWO FUNCTIONS
AS WELL AS HOW TO DETERMINE THE 
DERIVATIVES OF THESE FUNCTIONS.
THEY'RE BOTH 
COMPOSITE FUNCTIONS.
BUT FOR THIS FIRST FUNCTION 
HERE, THE EXPONENT OF 4
IS ATTACHED TO THE ENTIRE 
FUNCTION NATURAL LOG X,
AND THIS FUNCTION HERE, 
THE EXPONENT OF 4
IS ONLY ATTACHED TO X.
SO WE WILL HAVE TO APPLY 
THE CHAIN RULE
TO DIFFERENTIATE BOTH 
OF THESE FUNCTIONS,
AND THE KEY IS IDENTIFYING 
THE INNER FUNCTION
AND LETTING THAT EQUAL U.
SO FOR THIS FUNCTION, THE INNER 
FUNCTION IS NATURAL LOG X.
SO WE'LL LET U EQUAL 
NATURAL LOG X,
THEREFORE U PRIME OR DUDX 
IS GOING TO BE 1 DIVIDED BY X.
SO NOW, WE CAN THINK OF THIS 
FUNCTION AS U TO THE 4th
AND APPLY THE EXTENDED POWER 
RULE.
SO WE'D HAVE F PRIME OF X IS 
EQUAL TO THE DERIVATIVE OF U
TO THE 4th RESPECTS TO X,
THAT WOULD BE 4 x U TO THE 3rd, 
BUT U IS NATURAL LOG X x U PRIME
AND U PRIME IS 1/X.
SO HERE OUR DERIVATIVE FUNCTION 
IS GOING TO BE EQUAL TO 4
x NATURAL LOG RAISED TO THE 3rd 
POWER DIVIDED BY X.
NOW FOR OUR SECOND FUNCTION,
U IS GOING TO BE EQUAL TO X 
TO THE 4th,
THEREFORE DUDX OR U PRIME IS 
GOING TO BE EQUAL TO 4X CUBED.
SO WHEN WE WRITE THIS 
IN TERMS OF U,
IT'S GOING TO BE NATURAL LOG U.
SO NOW, WE'LL DIFFERENTIATE 
NATURAL LOG U WITH RESPECTS TO X
AND THE FORMULA IS HERE.
SO WE'LL HAVE F PRIME OF X 
IS EQUAL TO 1/U x U PRIME
WHICH WILL BE 1/X TO THE 4th x 
4X TO THE 3rd WHICH WOULD BE /1.
AND NOW, WE CAN SIMPLIFY 
3 FACTORS OF X OUT HERE.
THIS WILL SIMPLIFY TO 1, THIS 
WILL SIMPLIFY TO X TO THE 1st,
SO WE HAVE F PRIME OF X IS EQUAL 
TO 4 DIVIDED BY X.
NOW, THERE'S ONE MORE THING 
I WANT TO MENTION
ABOUT THIS FUNCTION HERE.
WE COULD APPLY THE POWER 
PROPERTY OF LOGARITHMS
TO THIS FUNCTION,
BUT NOT TO THE FUNCTION HERE 
ON THE LEFT.
F OF X = NATURAL LOG X TO THE 
4th IS EQUAL TO 4 NATURAL LOG X.
SO IN THIS FORM, F PRIME OF X 
WOULD JUST BE EQUAL TO 4
x THE DERIVATIVE OF NATURAL LOG 
X WHICH IS 1/X
WHICH DOES GIVE US THE SAME 
DERIVATIVE FUNCTION OF 4/X
AS WE SEE HERE.
BUT WE CANNOT APPLY 
THE POWER PROPERTY OF LOGARITHMS
TO THIS FUNCTION HERE,
BECAUSE OF RAISING THE ENTIRE 
FUNCTION OF THE 4th POWER
NOT JUST X AS WE HAVE HERE.
SO I HOPE THIS EXAMPLE HELPS 
CLARIFY THE DIFFERENCE
BETWEEN THESE TWO FUNCTIONS 
AS WELL AS THEIR DERIVATIVES.
NEXT, WE'LL TAKE A LOOK 
AT AN EXAMPLE
THAT REQUIRES THE PRODUCT RULE.
