# Graphical Intro to Derivatives and Integrals

Derivatives and Integrals have a two-way relationship!

Let's start by looking at sums and slopes:

### 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

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

Or, walking at 4 km per hour for 1 hour increases the distance by 4 km

Speed is the rate of change of distance

Change in distance is the sum of the speed over time

It will make more sense when you play with it below: change the distance line, or the speed line, to see its affect on the other:

images/deriv-integ.js?mode=1&topic=walking

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:

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

Many things have that same two-way relationship:

• Wealth and income
• Volume and flow rate
• Energy and power
• lots more!

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

images/deriv-integ.js?mode=multi

Integrals and Derivatives also have that two-way relationship!

Try it below, but first note:

• Δx (the gap between x values) only gives an approximate answer
• dx (when Δx approaches zero) gives the actual derivative and integral*
images/deriv-integ.js?mode=fn

*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