Finding the tangent and normal lines to f at x = x_0
You can change the function f (plotted in green) on the left-hand side. The derivative (shown also on the left) is calculated automatically and is plotted in blue.
It is possible to move the coordinate plane and to zoom in (e.g. by using the scroll wheel on your mouse).
You can change x_0 by using the slider on the top right of the GeoGebra window. It is the x-coordinate of the (black) point on f, in which the tangent line t (in red) and normal line n (in yellow) to f are plotted. Their expressions are shown on the right-hand side.
The key issue here:
In order to calculate t and n to f in x_0 you need the slope of f in x_0, which is f’(x_0). It is the value of the derivative of f in x_0. Instead of calculating this value, one could also look it up on the blue graph f’, on which the relevant point is marked as a blue point.