Graphical Intro to
Derivatives and Integrals

Derivatives and Integrals have a two-way relationship!

Let's start by looking at sums and slopes:

sage walks

Example: walking in a straight line

  • Walk slow, the distance increases slowly
  • Walk fast, the distance increases fast
  • Stand still and the distance won't change
  • Walk backwards, and you get closer to the start!

Walking at 4 km per hour for 1 hour makes the diistance increase by 4 km

A distance increase of 4 km in 1 hour gives a speed of 4 km per hour

Change in distance is the sum of the speed over time

Speed is the rate of change (slope) of distance

It will make more sense if you play with it below (drag the distance line at the top or the speed line below to see the other change):


Play with that a little and get comfortable with the two-way relationship. Try zero speed, or negative speed.

The slope of the distance line gives us the speed line, like this:

speed from distance

The "area" under the speed line gives us the increase in distance, like this:

distance from speed

Many things have that same two-way relationship:

Here is the same app as above, but you can choose different topics:


Integrals and Derivatives also have that two-way relationship!

Try it below, but first note:


*Note: this is a computer model and actually uses a very small Δx to simulate dx, and can make erors.

For true derivatives refer to Derivative Rules, and for integrals refer to Introduction to Integration