The Tangent Slope and the Derivative Y-Value
INSTRUCTIONS:
This applet will let you move point A on a function to examine the y-value of the derivative function.
1) Click on the move tool 
and then click on point A on the green curve f(x) to move the point manually. Examine how the tangent line slope "m" corresponds to the y-value of the red curve f'(x).
2) On the left input bar, you can animate the process by clicking on the "play button" arrow next to the coordinate point values of point A (you may need to scroll up).
3) Scroll down the input bar, click on the open circles of Point F, Point G, segment k and the tangent slope m1 to get the tangent line of f ' -- red curve -- to appear. Move point A again to see how the tangent line slope of "m1" corresponds to the y-value of the blue curve f ''(x).
and then click on point A on the green curve f(x) to move the point manually. Examine how the tangent line slope "m" corresponds to the y-value of the red curve f'(x).
2) On the left input bar, you can animate the process by clicking on the "play button" arrow next to the coordinate point values of point A (you may need to scroll up).
3) Scroll down the input bar, click on the open circles of Point F, Point G, segment k and the tangent slope m1 to get the tangent line of f ' -- red curve -- to appear. Move point A again to see how the tangent line slope of "m1" corresponds to the y-value of the blue curve f ''(x).Observation 1: Describe the relationship between the tangent line slope "m" for the segment BC at point A on the green curve f(x) and the y-value of point E on the red curve of f '(x). List 2 different specific x-value with tangent slopes "m" plus the corresponding coordinate points for E.
Observation 2: Describe the relationship between the tangent line slope "m1" (see instruction step 3) for the segment FG at point E on the red curve f'(x) and the y-value of point D on the blue curve of f ''(x). List 2 different specific x-values with tangent slopes "m1" plus the corresponding coordinate points for D.