- WE WANT TO DETERMINE 
THE DERIVATIVE
OF THE GIVEN FUNCTION 
F OF X = 2X SQUARED E
TO THE POWER OF 5X.
SO WE NEED TO RECOGNIZE 
HERE THAT WE HAVE A PRODUCT
OF TWO FUNCTIONS.
THE FIRST FUNCTION 
WE'LL LET EQUAL 2X SQUARED
AND THE SECOND FUNCTION 
WE'LL BE E TO THE POWER OF 5X.
SO WE'LL LET F = 2X SQUARED 
AND THEN WE'LL LET G = E
TO THE POWER OF 5X.
AND NOW WE'LL SET UP 
THE PRODUCT RULE GIVEN HERE.
SO F PRIME OF X IS GOING 
TO BE EQUAL
TO F x G PRIME + G x F PRIME.
OR THE FIRST FUNCTION 
2X SQUARED x THE DERIVATIVE
OF E TO THE POWER OF 5X 
+ THE SECOND FUNCTION E
TO THE 5X x THE DERIVATIVE 
OF THE FIRST FUNCTION
2X SQUARED.
NOTICE HOW I DIDN'T DETERMINE 
ANY DERIVATIVES YET.
I JUST SET THIS UP. 
SO NOW WE'LL GO BACK
AND DETERMINE THIS DERIVATIVE 
HERE AND THIS DERIVATIVE HERE.
WELL, FOR THE DERIVATIVE OF E 
TO THE POWER OF 5X
THIS IS A COMPOSITE FUNCTION, 
SO WE'LL LET U = 5X.
AND THE DERIVATIVE 
OF E TO THE U
IS GOING TO BE E 
TO THE U x U PRIME.
SO WE'LL HAVE x E 
TO THE 5X x U PRIME,
WHICH IS 5
+ E TO THE POWER OF 5X 
x THE DERIVATIVE OF 2X SQUARED,
WHICH WOULD BE 4X.
SO LET'S GO AHEAD 
AND DETERMINE THESE PRODUCTS.
HERE WE'LL HAVE 
10X SQUARED U TO THE 5X.
PLUS HERE WE'LL HAVE 
4X x E TO THE 5X.
ON THESE TYPES OF PROBLEMS 
YOU'LL NORMALLY SEE
THE FINAL DERIVATIVE 
IN FACTORED FORM.
SO THE GREATEST COMMON FACTOR 
WOULD HAVE A FACTOR OF 2,
THE FACTOR OF X, AND A FACTOR 
OF E TO THE POWER OF 5X.
SO IF WE FACTOR OUT 2X E 
TO THE 5X,
LET'S SEE WHAT'S LEFT.
WE'D HAVE A FACTOR OF 5X 
LEFT HERE
AND WE'D HAVE 
A FACTOR OF 2 HERE.
SO WHEN APPLYING 
THE PRODUCT RULE LIKE THIS
I THINK IT IS HELPFUL TO SHOW 
THIS STEP HERE
AND THEN DETERMINE 
THE DERIVATIVES
IN THE NEXT STEP.
WE'LL TAKE A LOOK AT AN EXAMPLE 
THAT INVOLVES
THE EXPONENTIAL FUNCTION 
AND THE QUOTIENT RULE
IN THE NEXT VIDEO.
