Derivative

The deriva­tive of \(y\) with re­spect to \(x\) de­scribes the rate at which \(y\) changes, given a change in \(x\). In par­tic­u­lar, we con­sider how tiny changes in one vari­able af­fect an­other vari­able. To take the deriva­tive of a func­tion, we can draw a line that is tan­gent to a graph of the func­tion. The slope of the tan­gent line is the value of the deriva­tive at that point. The deriva­tive of a func­tion \(f(x)\) is it­self a func­tion: it re­turns, for any \(x\), the slope of the line that is tan­gent to \(f(x)\) at the point \((x, f(x))\).

Examples

  • The time-deriva­tive of your car’s mileage is your car’s speed (be­cause your car’s speed is how quickly your car’s mileage changes over time)

  • The time-deriva­tive of your car’s speed is your ac­cel­er­a­tion (be­cause ac­cel­er­a­tion means how quickly your speed is chang­ing over time)

  • The time-deriva­tive of hu­man pop­u­la­tion size is the birth rate minus the death rate (be­cause if we take the birth rate minus the death rate, that tells how quickly the pop­u­la­tion is chang­ing over time)

  • The time-deriva­tive of wealth is in­come minus spend­ing (be­cause if we take in­come minus spend­ing, that tells us how your wealth is chang­ing over time)

  • The time-deriva­tive of blood pumped through your heart is the flow rate through the aor­tic valve (rates in gen­eral de­scribe how some­thing changes over time)

  • The time-deriva­tive of charge on a ca­pac­i­tor is the cur­rent flow­ing to it (cur­rents in gen­eral are time-deriva­tives of how much to­tal stuff has flowed)

  • The time-deriva­tive of how good of a life you’ll have lived is how happy you are right now (ac­cord­ing to he­do­nic util­i­tar­i­ans)

Okay, let’s take an­other stab at this. Time-deriva­tives, or deriva­tives “with re­spect to time,” de­scribe how things change over time. We can take deriva­tives with re­spect to other things too.

  • The deriva­tive of your car’s mileage with re­spect to how much fuel you’ve burned is your miles per gal­lon (be­cause your mpg de­scribes how your mileage changes per unit of fuel you burn)

  • The deriva­tive of the tem­per­a­ture of a pot of wa­ter with re­spect to how much heat you blast it with is called the heat ca­pac­ity of water

  • The deriva­tive of po­ten­tial en­ergy with re­spect to al­ti­tude is grav­i­ta­tional force (a higher up ob­ject has some stored en­ergy that it would re­lease if it fell; a lower down ob­ject has less “po­ten­tial en­ergy.” The change in po­ten­tial en­ergy as you change the al­ti­tude is why there’s a force in the first place.)

There were two goals with all those ex­am­ples, one ex­plicit, and one covert. The ex­plicit one was to give you a sense for what deriva­tives are. The covert one was to quietly sug­gest that you will never un­der­stand the the way the world works un­less you un­der­stand deriva­tives. But hey, look at you! You kind of un­der­stand deriva­tives already! Let’s get to the math now, shall we?

Set­ting Up The Math

You just got your new car.

It’s a Tesla be­cause you care about the en­vi­ron­ment al­most as much as you care about look­ing awe­some. Your mileage is sit­ting at 0. The world is your oys­ter. At time \(t = 0\), you put your foot on the ac­cel­er­a­tor. For the next few sec­onds, your mileage will be \(4.7 t^2\), where your mileage is in me­ters, and \(t\) is in sec­onds since you pressed your foot on the ac­cel­er­a­tor. Now the first ques­tion is this: if that equa­tion tells us how many me­ters we’ve trav­eled af­ter how many sec­onds, how fast are we go­ing at any given point in time?

The as­tute reader will have no­ticed that this was the first ex­am­ple of a time-deriva­tive that we gave: the time-deriva­tive of your car’s mileage is your car’s speed. Let’s think about this sans math for a sec­ond. If we know where we are at any time, we should be able to figure out how fast we’re go­ing. There isn’t any ex­tra in­for­ma­tion we need. The only ques­tion is how. Well, we take the deriva­tive of the mileage with re­spect to time to get our speed. In other words:

$$\frac{\mathrm{d}}{\mathrm{d} t} mileage = speed$$

That means the deriva­tive with re­spect to \(t\), where \(t\) is the time in sec­onds. But we know what the mileage is, in terms of \(t\). Our mileage is just \(4.7 t^2\). So we can write:

$$\frac{\mathrm{d}}{\mathrm{d} t} 4.7 t^2 = speed$$

Solv­ing The Math

Sorry to leave you hang­ing for a sec, but we’re go­ing to start with some­thing a lit­tle sim­pler.

$$distance\ traveled = 2t$$

If this is the graph of how far some­one has trav­eled af­ter how many sec­onds, we can see that ev­ery sec­ond they go 2 more me­ters. In other words, they are trav­el­ing 2 me­ters per sec­ond, which you might no­tice is the slope of this line. In gen­eral, the deriva­tive of a func­tion is like the slope of the func­tion when you graph it out.

This works fine if our func­tion is some­thing like \(distance\ traveled = 2t\). What if our func­tion isn’t a line though. What if it’s \(distance\ traveled = t^2\)?

Things that aren’t lines don’t have slopes. So if this is a graph of our dis­tance trav­eled over time, it’s not as easy to see how fast we were go­ing. But let’s say we want to see how fast we were go­ing at \(t=1\). If we zoom in enough on that curve, it will start to flat­ten out into a straight line un­til we can’t tell the differ­ence. The slope of that line is what gives us our speed. The pro­cess of tak­ing a curve like this one, and get­ting the “slope” at any given point is called “tak­ing the deriva­tive.”

Let’s take the deriva­tive of \(d = t^2\), where \(d\) is the dis­tance and \(t\) is the time. (We’ll take the deriva­tive with re­spect to \(t\)). Pre­pare your­self, now take a look at the graph down there.

We know how to find the slope of a line if we’re given two points, so we’re go­ing to do that, and then slowly move the points to­gether un­til they’re on top of each other. The co­or­di­nates of the points are shown above, and we can calcu­late the slope pretty eas­ily by do­ing \(\frac{\Delta d}{\Delta t}\). This gives us a slope of 2.

Now let’s say that our first point is at \((t,t^2)\), that our sec­ond point is \(h\) units to the right, so it’s co­or­di­nates are \(((t+h),(t+h)^2)\). Now we have:

$$∆d=(t+h)^2-t^2$$
$$∆t=(t+h) - t$$
Alge­bradabra:
$$∆d=2ht + h^2$$
$$∆t=h$$
$$\frac{\Delta d}{\Delta t}=\frac{2ht + h^2}{h}=2t+h$$

Now as we make \(h\) re­ally small, the points get closer and closer to­gether, and the slope of the line be­comes \(2t\). So when \(t\) is \(1\), the slope is \(2\). And when \(t\) is \(5\), the slope is \(10\).

We say the deriva­tive of \(t^2\) is \(2t\). With similar logic, you can show that the deriva­tive of \(4.7t^2\) is \(9.4t\). And that means that if you put your foot on the ac­cel­er­a­tor of your new Tesla at time \(t=0\), your speed af­ter \(t\) sec­onds will be \(9.4t\). After 1 sec­ond, you’ll be trav­el­ing 9.4 me­ters per sec­ond. After 3 sec­onds, you’ll be go­ing 28.2 me­ters per sec­ond (or 64 mph).

Concluding

That’s what deriva­tives are. The Tesla case was just one ex­am­ple of ac­tu­ally find­ing the deriva­tive of some­thing. Ob­vi­ously, if our dis­tance trav­eled had been some to­tally differ­ent func­tion of time, the deriva­tive would have been differ­ent. We found that the deriva­tive of \(t^2\) is \(2t\). Below is a list of other deriva­tives. You can imag­ine that if the func­tion on the left was our dis­tance trav­eled af­ter a time \(t\), the func­tion on the right would be our speed at a time \(t\). (All of the \(c\)‘s and \(n\)’s are con­stants.)

$$\frac{\mathrm{d} }{\mathrm{d} t}c=0$$
$$\frac{\mathrm{d} }{\mathrm{d} t}ct=c$$
$$\frac{\mathrm{d} }{\mathrm{d} t}ct^2=2ct$$
$$\frac{\mathrm{d} }{\mathrm{d} t}ct^2=3ct^2$$
$$\frac{\mathrm{d} }{\mathrm{d} t}ct^n=nct^{n-1}$$
$$\frac{\mathrm{d} }{\mathrm{d} t}e^t=e^t$$
$$\frac{\mathrm{d} }{\mathrm{d} t}sin(t)=cos(t)$$
$$\frac{\mathrm{d} }{\mathrm{d} t}cos(t)=-sin(t)$$

If you’re up for it, try to use the method we showed for solv­ing deriva­tives to ver­ify some of these. Good luck!

See also

If you en­joyed this ex­pla­na­tion, con­sider ex­plor­ing some of Ar­bital’s other fea­tured con­tent!

Ar­bital is made by peo­ple like you, if you think you can ex­plain a math­e­mat­i­cal con­cept then con­sider con­tribut­ing to ar­bital!

Parents:

  • Mathematics

    Math­e­mat­ics is the study of num­bers and other ideal ob­jects that can be de­scribed by ax­ioms.