- WE WANT TO USE THE GRAPH 
OF F OF X
TO ESTIMATE THE VALUE OF F 
PRIME OF 3
OR THE VALUE OF A DERIVATIVE 
FUNCTION AT X = 3.
F PRIME OF 3 = THE SLOPE 
OF THE TANGENT LINE AT X = 3.
IT ALSO GIVES US 
THE INSTANTANEOUS RATE OF CHANGE
OF THE FUNCTION AT X = 3.
SO FOR THE FIRST STEP WE'LL 
LOCATE THE POINT ON THE FUNCTION
WHEN X = 3, 
WHICH WOULD BE THIS POINT HERE.
NEXT WE'LL SKETCH 
THE TANGENT LINE AT THIS POINT
AND THEN FIND THE SLOPE 
OF THAT LINE.
SO THIS RED LINE 
IS OUR TANGENT LINE AT X = 3
AND NOW WE'LL LOCATE TWO POINTS 
ON THIS LINE
WITH INTEGER COORDINATES 
TO FIND THE SLOPE OF THE LINE.
SO LET'S GO AHEAD AND USE THIS 
POINT OF TANGENCY HERE
WITH COORDINATES (3,2)
AND LET'S ALSO USE THIS POINT 
HERE WITH COORDINATES (5,4).
AND NOW WE'LL FIND THE SLOPE 
OF THIS LINE
WHERE THE SLOPE = THE CHANGE 
OF Y DIVIDED BY THE CHANGE OF X.
SO WE COULD FIND THE SLOPE 
USING THESE TWO POINTS
IN THIS FORMULA HERE.
BUT WE CAN ALSO FIND THIS SLOPE
BY ANALYZING 
THE COORDINATE PLANE.
NOTICE TO MOVE FROM THIS POINT 
ON THE LEFT
TO THIS POINT ON THE RIGHT 
WE'D HAVE TO GO UP TWO UNITS,
WHICH MEANS THE CHANGE OF Y 
IS +2.
AND THEN WE HAVE TO GO 
RIGHT TWO UNITS,
WHICH MEANS THE CHANGE OF X 
IS ALSO +2.
WHICH MEANS F PRIME OF 3, 
WHICH AGAIN = THE CHANGE OF Y
DIVIDED BY THE CHANGE OF X 
WOULD BE = 2 DIVIDED BY 2 OR +1.
BUT LET'S ALSO SHOW HOW WE CAN 
DO THIS USING THE COORDINATES
WHERE WE'LL CALL THIS 
(X SUB 1,Y SUB 1)
AND WE'LL CALL THESE COORDINATES 
(X SUB 2,Y SUB 2).
SO USING THE COORDINATES 
WE'D HAVE F PRIME OF 3
= Y SUB 2 - Y SUB 1,
THAT WOULD BE 4 - 2 DIVIDED BY X 
SUB 2 - X SUB 1,
THAT WOULD BE 5 - 3.
GIVING US THE SAME RESULT OF 2 
DIVIDED BY 2 OR +1.
I HOPE YOU HAVE FOUND 
THIS HELPFUL.
