Geometry related to derivatives
Derivative equals speed
Here we assume that a point moves along a straight line and try to find its speed as usual. We divide the distance traveled by the time it takes to travel it. The ratio is an average speed over a time interval. As we reduce the length of the time interval, we get a better and better approximation to the exact (instantaneous) speed at a point in time.
Position of point as a function of time
Working with the visualization of the point movement on a straight line is inconvenient because it is difficult to correlate the point position to time. It is much better to visualize the movement on the space-time plane where the horizontal axis is for time and the vertical axis is for the point position.
Video 2. Position of point as function of time
Measuring the slope of a straight line
A little digression: how do you measure the slope of a straight line, if you know the values of the function at different points?
Derivative as the slope of a tangent line
This is like putting two and two together: we apply the previous definition to the slope of a secant drawn through two points on a graph. Then it remains to notice that the secant approaches the tangent line, as the second point approaches the first.
From function to its derivative
This is a very useful exercise that allows later to come up with the optimization conditions, called first order and second order conditions.
Let be some function and fix an initial point . The derivative is defined as the limit
When describes the movement of a point along a straight line, the derivative gives the speed of that point. When is drawn on a plane, the derivative gives the slope of the tangent line to the graph.