- I WANT TO TAKE A LOOK 
AT TWO EXAMPLES
THAT SOMETIMES CAUSE PROBLEMS 
FOR STUDENTS.
HERE WE HAVE F OF X = SINE 
TO THE 4th X.
AND HERE WE HAVE THE FUNCTION 
F OF X = SINE X TO THE 4th.
THESE ARE TWO ENTIRELY 
DIFFERENT FUNCTIONS
WITH ENTIRELY 
DIFFERENT DERIVATIVES.
I THINK IT'S A LITTLE BIT 
MORE OBVIOUS
WHEN THERE ARE PARENTHESIS HERE.
BUT SOMETIMES SOME TEXTBOOKS 
LEAVE OFF THE PARENTHESIS
WHICH CAN MAKE IT EVEN MORE 
DIFFICULT TO DISTINGUISH
BETWEEN THESE TWO FUNCTIONS.
SO LET'S GO AHEAD 
AND TAKE A LOOK
AT THIS FIRST FUNCTION HERE, 
F OF X = SINE TO THE 4th X.
THE FIRST THING WE NEED 
TO RECOGNIZE HERE
IS THAT THIS IS THE SAME AS 
SINE X RAISED TO THE 4th POWER.
SO WE DO HAVE 
A COMPOSITE FUNCTION,
AND THEREFORE WE'LL HAVE 
TO APPLY THE CHAIN RULE.
SO IN THESE TYPES OF PROBLEMS,
THE MOST IMPORTANT THING IS 
TO IDENTIFY THE INNER FUNCTION
AND LET THE INNER FUNCTION 
EQUAL U.
SO FOR THIS FUNCTION, SINE X 
IS GOING TO BE EQUAL TO U.
AND LET'S RECORD THAT OVER HERE.
WHICH MEANS WE CAN NOW 
REWRITE THE FUNCTION
IN TERMS OF U AS U TO THE 4th.
SO NOW WE CAN APPLY 
THE EXTENDED POWER RULE
THAT HAS THE CHAIN RULE BUILT IN 
TO DETERMINE F PRIME OF X.
SO F PRIME OF X IS GOING TO BE 
EQUAL TO THE DERIVATIVE OF
RESPECTS TO U TIMES U PRIME.
SO N x U TO THE N - 1 
WOULD BE THE DERIVATIVE
OF THE OUTER FUNCTION,
AND U PRIME WOULD BE 
THE DERIVATIVE
OF THE INNER FUNCTION.
SO WE'D HAVE 4U TO THE 3rd TIMES 
U PRIME AS OUR DERIVATIVE.
BUT WE WANT OUR DERIVATIVE 
IN TERMS OF X.
SO WE'LL SUBSTITUTE SINE X 
FOR U,
AND WE'LL HAVE TO DETERMINE 
U PRIME.
WELL, U PRIME WOULD BE 
THE DERIVATIVE OF SINE X
WHICH IS COSINE X.
SO WE'LL HAVE F PRIME OF X = 4 
x SINE X TO THE 3rd,
AND THEN U PRIME = COSINE X.
LET'S GO AHEAD AND WRITE THIS 
ONE MORE TIME.
WE HAVE F PRIME OF X = WE HAVE 
4 SINE TO THE 3rd X COSINE X.
WE COULD EXPRESS THIS DERIVATIVE 
IN A DIFFERENT FORM
USING THE DOUBLE ANGLE IDENTITY 
FOR SINE,
BUT WE'RE GOING TO GO AHEAD 
AND LEAVE IT IN THIS FORM
FOR THIS EXAMPLE.
NOW, GIVEN -- LET'S GO AHEAD 
AND COMPARE THIS
TO THE DERIVATIVE SINE OF X 
TO THE 4th.
THIS IS A COMPOSITE FUNCTION.
SO THE INNER FUNCTION HERE 
WOULD BE X TO THE 4th.
SO IF WE LET U - X TO THE 4th WE 
CAN REWRITE THE GIVEN FUNCTION
IN TERMS OF U AS JUST SINE U.
AND NOW TO DETERMINE F PRIME 
OF X
WE'LL APPLY THE DERIVATIVE 
FUNCTION FOR SINE U
WHICH INCLUDES THE CHAIN RULE 
AS WE'LL SEE HERE.
SO WE HAVE THE DERIVATIVE 
OF SINE U WITH RESPECTS TO U,
THAT'S GOING TO BE COSINE U 
x U PRIME.
SO WE HAVE THE DERIVATIVE 
OF THE OUTER FUNCTION
TIMES THE DERIVATIVE 
OF THE INNER FUNCTION.
SO NOTICE HOW WE WILL HAVE 
TO DETERMINE U PRIME.
WELL, U PRIME WOULD BE 4X 
TO THE 3rd.
SO NOW WE'LL REWRITE 
OUR DERIVATIVE FUNCTION
IN TERMS OF X RATHER THAN U.
SO F PRIME OF X IS GOING 
TO BE EQUAL TO COSINE U,
BUT U IS X TO THE 4th, 
AND U PRIME IS 4X CUBED.
LET'S GO AHEAD 
AND REARRANGE THESE TERMS.
LET'S WRITE IT AS 4X CUBED 
COSINE X TO THE 4th.
SO IT IS IMPORTANT 
THAT WE CAN DISTINGUISH
BETWEEN THE TWO DIFFERENT 
FUNCTIONS GIVEN IN THIS VIDEO.
I HOPE YOU FOUND THESE EXAMPLES 
HELPFUL.
