Graphing the Derivative from Any Function

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  • 0:10 Location as a Function…
  • 1:01 The Derivatives of a Function
  • 3:46 Derivative of a Single…
  • 7:40 Rules of Derivation
  • 8:08 Discontinuous Derivatives
  • 11:08 Applications to Other…
  • 14:14 Lesson Summary
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Lesson Transcript
Instructor: Heather Higinbotham
When you know the rules, calculating the derivates of equations is relatively straightforward, although it can be tedious! What happens, though, when you don't know the function? In this lesson, learn how to graph the derivative of a function based solely on a graph of the function!

Location as a Function of Time

I often have this dream where I'm back in high school, and I've just learned how to drive. My father's there, and he doesn't exactly trust me in the car, so he puts a GPS tracker on my car. He was going to sit at his computer and watch exactly where I was at any given point in time. Let's say that this GPS tracker measured the distance I was, at any given point in time, from home. So for example, one day he's sitting at his computer, and he sees this graph of my location as a function of time. Over here, he sees that I left home, I hit a traffic jam, I went to the mall and then, eventually, I came home - a little bit late, he was sure to remind me.

The Derivatives of a Function

The slope of the line is very steep when speeding home, so the derivative is a large negative number
Derivatives of a Function Mall Example

Other than my location as a function of time, he was also interested in how fast I was going - my velocity. So he can use the fact that my velocity is the derivative of my position as a function of time to determine, approximately, how fast I was going at any given point in time. So dx/dt.

Right here, I've just left home, and to make it look good, I pull out of the driveway very, very slowly. My derivative, which is the slope of the tangent, is very small. It's positive because I'm leaving home, but it's small. As I get out of the house, I speed up. My position changes a lot faster as a function of time. That's true until I hit this traffic jam, which slows me down again. So up here, my derivative is smaller. When I get to the mall, I stop the car. When I park the car, its position is not changing as a function of time. So here, my derivative is flattened out. It looks like it's zero. When I'm done shopping, I start to head home. Now here the slope of the tangent is negative, because I'm on my way back home. On this particular day, I forgot something at the mall, so I had to turn around and go back to the mall. By that time I was really late, so I sped home. My derivative is very steep here; the slope of this line is very steep. I'm driving very quickly because I'm very late. So the derivative here is a very large negative number.

So let's say my father wants to know exactly how fast I'm going, and let's say he zooms in really close to one part of this map. To find out exactly how fast I was going, he draws a tangent on this line, and he finds the slope of this tangent. In this case, my slope is dx/dt or the change in x divided by the change in time. Now as x went from 348 to 349, the change is 1. In this case, it is 1 foot. The change in time was 1 second, going from t=1 to t=2. So at this point in time, I was traveling at a whopping 1 foot per second, which is 60 feet per minute, which is not all that fast.

Derivative of a Single Destination

So let's get rid of all of the numbers here, and just look at one example of my position as a function of time. And let's see what my father can glean from this single graph. This graph is pretty simple. I leave home, I get to the mall, I turn around and I come back home. So let's focus on this area where I'm leaving home. I'm going away from home. Because I'm leaving home, I know that my derivative is going to be positive. It's going to be greater than zero for this entire time where I'm leaving home. So this is where x is increasing as t increases, but let's be a little more specific. Let's say in that first minute, I sped off very quickly. I've gone a long way in just one minute. In this case, the slope of the tangent is pretty large and positive. Once I get further from home, I slow down a little bit. I'm not moving - I'm not getting quite as far - as a function of time. In one minute, maybe I go half the distance that I went before. Here, the slope of the tangent isn't quite as steep as it was when I first left home; I'm not moving quite as fast. When I get even further from home, I slow down even more. So here in one minute, I'm hardly moving. My derivative is still positive, but it's pretty close to zero because x is not changing very much as a function of time.

So coming home, it's like the same thing in reverse, like I'm driving backward. Now my derivative is going to be negative. How fast I'm going will determine how negative it is. Is the value small but negative, or is it large but negative? If it's small but negative, I'm going backward slowly. And if it's large and negative, I'm going backward very quickly. When I first leave the mall, my position is not changing very much as a function of time. My derivative is negative, but it's close to zero. As I get closer to home, I speed up some. So my derivative is still negative, it's just slightly larger. It's further away from the x-axis. As I get even closer to home, I realize that I'm very late, and I've sped up a lot. And so the magnitude of my velocity is very large, but it's still negative because I'm still driving backward to get home.

Let's put all of this together, and let's take our position as a function of time and use it to graph our velocity, our x`, as a function of time. So here when we're leaving home, we started off with a really fast velocity, and positive because we were going to the mall forward. As we got to the mall, we slowed down. Our derivative decreased. When we got to the mall, we actually stopped for a second. Our derivative, right here, is actually equal to zero. The slope of the tangent here is zero. Then we started to come back from the mall; we started driving backward. At first we were driving slowly. Our slope was very shallow, so our derivative was negative but close to zero. Then we started to speed up, getting more and more and more negative. By the time we got home, we were going very fast, but we were still going backward. So our derivative is down here. We can connect all of these points and come up with an approximation for x`. And it looks like this straight line that goes through zero when we were stopped at the mall.

Graphs depicting the derivative of the mall trip
Mall Trip Derivative Graphs

Rules of Derivation

Let's take a second to make this a little more formal. Let's look at x as a function of time. When our function x is getting larger, we're leaving home. We have a positive derivative. The derivative x` is going to be greater than zero. If the function is getting smaller (we're driving backwards here), we have a negative derivative. If our function isn't changing, the derivative will be zero. This is like that instant when we were at the mall.

Discontinuous Derivatives

I got into a little bit of trouble for my last exploit. My father didn't let me out of the house for a while, but when he did, I again went to the mall. And he plotted my position as a function of time, and it looked like this. Now here when I left the house, I kept a pretty constant velocity. So all along this time, the slope of the tangent, everywhere along here, was the same because this is a straight line. I got to the mall and I stayed there for a while, and then I came home. And again, I had a straight line between the mall and home.

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