- NOW WE'LL LOOK AT TWO EXAMPLES 
OF DETERMINING
DERIVATIVE FUNCTION VALUES 
GIVEN TWO BASIC FUNCTIONS.
FIRST WE HAVE IF F OF X 
= 2X - 3,
THEN F PRIME OF 3 IS EQUAL 
TO WHAT VALUE?
SO BEFORE WE DETERMINE 
F PRIME OF 3,
WE NEED TO FIND 
OUR DERIVATIVE FUNCTION.
SO IF F OF X = 2X - 3,
THEN F PRIME OF X IS GOING TO 
= 2 x THE DERIVATIVE OF X.
WELL, THE DERIVATIVE OF X 
IS ALWAYS EQUAL TO 1,
SO THAT WILL JUST GIVE US 2 X 1 
OR 2 - THE DERIVATIVE OF 3.
THE DERIVATIVE OF ANY CONSTANT 
IS ALWAYS 0
WHICH MEANS 
OUR DERIVATIVE FUNCTION
IS THE CONSTANT FUNCTION 
F PRIME OF X = 2.
AND NOTICE HOW THIS DERIVATIVE 
FUNCTION VALUE
IS NOT AFFECTED BY X
THEREFORE F PRIME OF 3 IS GOING 
TO BE EQUAL TO 2.
THE VALUE OF THE DERIVATIVE 
FUNCTION AT A GIVEN VALUE OF X
GIVES US THE SLOPE OF THE 
TANGENT LINE AT THAT LOCATION.
SO LET'S TAKE A LOOK AT 
WHAT'S HAPPENING GRAPHICALLY.
IF WE GRAPH THE ORIGINAL 
FUNCTION F OF X = 2X - 3,
WE HAVE THIS LINE HERE THAT 
HAS A Y-INTERCEPT OF -3
AND A SLOPE OF 2.
FROM HERE WE CAN GO UP 2 
AND RIGHT 1.
NOW, IF WE LOCATE THE POINT ON 
THE FUNCTION WHEN X = 3
WE'D BE AT THIS POINT HERE,
AND IF WE TRY TO SKETCH 
A TANGENT LINE
TO OUR GRAPH AT THIS POINT
WE CAN SEE OUR TANGENT LINE 
PASSING THROUGH THIS POINT
WOULD BE THE SAME LINE 
AS THE ORIGINAL FUNCTION
AND THEREFORE THE SLOPE 
OF THE TANGENT LINE
WOULD BE THE SAME SLOPE 
AS THE ORIGINAL FUNCTION
WHICH WE KNOW IS 2.
THIS IS WHY ANY POINT 
ON THIS LINE,
THE DERIVATIVE FUNCTION VALUE 
WOULD ALWAYS BE 2.
NOW, LET'S TAKE A LOOK 
AT OUR SECOND EXAMPLE.
IF G OF X EQUALS THE CONSTANT 7,
THEN G PRIME OF -2 
IS EQUAL TO WHAT VALUE?
SO AGAIN, WE'LL FIRST FIND 
OUR DERIVATIVE FUNCTION,
SO G OF X = 7.
G PRIME OF X IS ALWAYS GOING 
TO BE EQUAL TO 0,
AND WITH A DERIVATIVE 
OF A CONSTANT IS ALWAYS 0.
AND AGAIN,
THIS DERIVATIVE FUNCTION VALUE 
IS NOT AFFECTED BY X
THEREFORE G PRIME OF -2 
IS ALSO EQUAL TO 0.
SO THIS IS TELLING US THAT IF 
WE TRY TO SKETCH A TANGENT LINE
TO OUR CONSTANT FUNCTION 
AT X = -2
THE TANGENT LINE WILL HAVE 
A SLOPE OF 0.
LET'S LOOK AT THIS GRAPHICALLY 
AS WELL.
HERE'S A GRAPH OF G OF X = 7.
IT'S THIS HORIZONTAL LINE 
PASSING THROUGH 7 ON THE Y AXIS.
HERE'S THE POINT ON THE FUNCTION 
WHEN X = -2.
REMEMBER, THE SLOPE OF ANY 
TANGENT LINE IS EQUAL TO 0,
AND IF WE TRY TO SKETCH 
A TANGENT LINE
TO OUR HORIZONTAL LINE 
AT THIS POINT HERE,
IT'S GOING TO BE THE SAME 
HORIZONTAL LINE,
AND THEREFORE THE SLOPE = 0.
AND THIS WOULD BE THE CASE
FOR ANY POINT THAT WE SELECTED 
ON THIS HORIZONTAL LINE.
AND THAT'S WHY THE DERIVATIVE 
FUNCTION FOR A HORIZONTAL LINE
IS ALWAYS EQUAL TO 0.
OKAY, I HOPE YOU FOUND 
THIS EXPLANATION HELPFUL.
