What is a derivative?

Derivatives can be a tough concept to grasp, so we're going to start out first with understanding them visually / graphically with a real-world example. Then, we'll dive into the nitty-gritty of learning how to solve for them mathematically!

Derivatives explained

Let’s say you’re driving to Coachella, and as you get closer and closer to the concert, the traffic gets worse and worse.

Imagine that the below graph represents your distance traveled throughout your drive to Coachella:

Now, picture this: you arrive at your destination, and you're asked the following question:

What was your average speed on your drive here?

To solve for this, you'd simply find the total distance...

...and divide it by the total time.

This would result in the following average line...

...with a slope of 4mph representing your average speed throughout your drive to Coachella.

Easy enough... but what if you were asked this instead:

What was your speed at exactly 2hrs into the drive?

Well... that one would be a little tougher to determine with just this graph. We know our total distance traveled at 2hrs was around 1.3 miles...

...but to determine our speed at that point, we'd need to determine how our total distance function was changing at exactly 2hrs into the drive. We'd basically need to find the slope at that given point.


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