- WE WANT TO USE THE GRAPH 
OF F OF X
TO ESTIMATE THE VALUE OF F 
PRIME OF 4
OR THE VALUE OF THE DERIVATIVE 
FUNCTION WHEN X = 4.
F PRIME OF 4 = THE SLOPE 
OF THE TANGENT LINE AT X = 4.
IT'S ALSO THE INSTANTANEOUS RATE 
OF CHANGE
OR THE FUNCTION AT X = 4.
SO FOR THE FIRST STEP 
WE'LL LOCATE THE POINT
ON THE FUNCTION WHEN X = 4, 
WHICH WOULD BE HERE.
AND NOW WE'LL SKETCH 
THE TANGENT LINE AT THIS POINT
AND FIND THE SLOPE OF THE LINE.
SO THE RED LINE 
IS OUR TANGENT LINE
AND NOW WE'LL FIND TWO POINTS 
ON THIS LINE
WITH THE INTEGER COORDINATES 
TO FIND THE SLOPE OF THE LINE.
SO WE CAN GO AHEAD AND USE 
THE POINT OF TANGENCY HERE
WITH COORDINATES (4,3)
AND LET'S USE THIS POINT HERE 
THAT HAS COORDINATES (5,1).
AND NOW TO FIND THE SLOPE 
OF THIS LINE
WE CAN USE OUR SLOPE FORMULA 
HERE,
WHERE THE SLOPE IS EQUAL 
TO THE CHANGE OF Y
DIVIDED BY THE CHANGE OF X.
BUT WE CAN ALSO FIND THE SLOPE 
USING THE COORDINATE PLANE.
IF WE WANT TO GO FROM THE POINT 
ON THE LEFT
TO THE POINT ON THE RIGHT
NOTICE HOW WE HAVE TO GO DOWN 
TWO UNITS,
WHICH MEANS THE CHANGE OF Y 
IS -2.
AND THEN WE'D HAVE TO GO RIGHT 1 
UNIT, SO THE CHANGE OF X IS +1.
WHICH MEANS F PRIME OF 4
WHICH IS EQUAL TO THE SLOPE 
OF OUR TANGENT LINE = -2
DIVIDED BY 1 OR JUST -2.
OUR OTHER OPTION IS TO USE 
THE SLOPE FORMULA HERE
AND I'LL ALSO SHOW THAT.
SO IF WE CALL THESE THE 1s, 
MEANING (X SUB 1,Y SUB 1)
AND THESE THE 2s, 
MEANING (X SUB 2,Y SUB 2)
USING OUR SLOPE FORMULA WE CAN 
ALSO DETERMINE F PRIME OF 4.
F PRIME OF 4 WOULD BE 
= (Y,2) - (Y,1),
THAT WOULD BE 1 - 3 
DIVIDED BY X SUB 2 - X SUB 1,
WHICH SHOULD BE 5 - 4,
GIVING US -2 DIVIDED BY 1 
OR THE SAME RESULT OF -2.
LET'S TAKE A MOMENT AND DISCUSS 
WHAT THIS ACTUALLY MEANS.
IF F PRIME OF 4 = -2/+1 WHICH IS 
EQUAL TO THE CHANGE OF Y
OR THE CHANGE IN THE FUNCTION 
VALUE
DIVIDED BY THE CHANGE OF X,
THIS TELLS US THAT WHEN X = 4 
THE FUNCTION VALUE IS CHANGING
AT A RATE OF -2 UNITS PER 1 UNIT 
CHANGE OF X.
I HOPE YOU HAVE FOUND 
THIS HELPFUL.
