- WE WANT TO FIND THE DERIVATIVE 
OF F OF X = THE QUANTITY 3X
SQUARED - 2X + 7 x E 
TO THE POWER OF X.
SO BECAUSE F OF X IS A PRODUCT 
OF 2 FUNCTIONS,
WE'LL HAVE TO APPLY THE PRODUCT 
RULE OF DIFFERENTIATION
IN ORDER TO FIND OUR DERIVATIVE.
THE PRODUCT RULE IS STATED HERE.
THE DERIVATIVE OF THE PRODUCT OF 
2 FUNCTIONS, F AND G,
IS EQUAL TO THE FIRST FUNCTION
x THE DERIVATIVE OF THE SECOND 
FUNCTION + THE SECOND FUNCTION
x THE DERIVATIVE OF THE FIRST 
FUNCTION.
SO WE'LL LET THE POLYNOMIAL 
FUNCTION BE FUNCTION F
AND THE EXPONENTIAL FUNCTION
BE FUNCTION G.
SO LET'S START 
BY SETTING UP THE PRODUCT RULE.
WE WOULD F PRIME OF X IS EQUAL 
TO THE FIRST FUNCTION
OR THE POLYNOMIAL FUNCTION
x THE DERIVATIVE OF THE SECOND 
FUNCTION
OR x THE DERIVATIVE OF E TO THE 
X + THE SECOND FUNCTION
WHICH IS E TO THE X x THE 
DERIVATIVE OF THE FIRST FUNCTION
WHICH IS THE POLYNOMIAL 
FUNCTION.
NOTICE HOW WE HAVEN'T FOUND ANY 
DERIVATIVES YET,
WE JUST SET UP THE PRODUCT RULE.
SO NOW, WE'LL FIND
THE DERIVATIVE HERE AND HERE,
THEN FIND THE PRODUCTS 
AND THEN SIMPLIFY.
SO F PRIME OF X IS GOING TO BE 
EQUAL TO 3X SQUARED - 2X + 7
x THE DERIVATIVE OF E TO THE X 
WHICH IS JUST E TO THE X + E
TO THE X x THE DERIVATIVE OF THE 
POLYNOMIAL FUNCTION.
SO TO FIND THE DERIVATIVE 
OF THIS POLYNOMIAL FUNCTION,
WE'LL APPLY THE POWER RULE OF 
DIFFERENTIATION GIVEN HERE.
SO WE'D HAVE 3 x THE DERIVATIVE 
OF X SQUARED, SO WE'RE GOING
TO MULTIPLY BY 2, THAT WOULD BE 
6, SUBTRACT 1 FROM THE EXPONENT,
THAT WOULD BE X TO THE 1st - 2 
x THE DERIVATIVE OF X.
THE DERIVATIVE OF X 
IS EQUAL TO 1,
SO WE JUST HAVE -2 
+ THE DERIVATIVE OF 7.
THE DERIVATIVE OF A CONSTANT 
IS EQUAL TO 0,
SO OUR DERIVATIVE 
IS JUST 6X - 2.
SO NOW, WE'LL GO AHEAD 
AND DISTRIBUTE HERE,
DISTRIBUTE HERE, 
AND COMBINE LIKE TERMS.
SO WE'LL HAVE 3 PRODUCTS HERE,
1, 2, 3, AND 2 PRODUCTS HERE.
SO IF PRIME OF X IS EQUAL TO 3X 
SQUARED E TO THE X - 2X
E TO THE X + 7 E TO THE X,
AND HERE WE'LL HAVE + 6X E 
TO THE X - 2 E TO THE X.
NOW, WE'LL GO AHEAD AND COMBINE 
LIKE TERMS.
THESE TWO TERMS ARE LIKE TERMS,
AND THESE TWO TERMS ARE LIKE 
TERMS.
SO WE HAVE F PRIME OF X = 3X 
SQUARED E TO THE X,
-2X E TO THE X + 6X E TO THE X 
WOULD BE + 4X E TO THE X,
AND THEN 7 E TO THE X - 2 E TO 
THE X WOULD BE + 5 E TO THE X.
AND NOW TO FINISH, 
WE COULD FACTOR THIS.
NOTICE HOW EACH TERM HAS A 
COMMON FACTOR OF E TO THE X,
SO F PRIME OF X IS EQUAL TO E TO 
THE X x THE QUANTITY 3X SQUARED
+ 4X + 5.
AND THIS SHOULD BE OUR 
DERIVATIVE FUNCTION.
I HOPE YOU FOUND THIS HELPFUL.
