Name: Derivatives: Crash Course Physics #2
Uploaded: 2019-03-25T12:48:17.000Z
Duration: 10 min 2 s
Description: Thousands of YouTube videos with English-Chinese subtitles! Now you can learn to understand native speakers, expand your vocabulary, and improve your pronunciation...

Adverbs of degree

Every discipline of science has its very own
special language --

the way it communicates the ideas that it investigates.

For example, biology finds order in the world,
by giving every living thing a name, in Latin.

Chemistry has a system of prefixes, suffixes,
and numerals to tell you, in a word or two,

the exact composition of an atom, or a compound.

But physics has to communicate its ideas differently.

Because, if you're trying to describe how
the world works, you really have to know how

things relate to each other in a
mathematical way.

For example, we've been talking a lot about position, velocity, and acceleration, and how they're all connected.

Velocity is a measure of your change in position, and acceleration is a measure of your  change in velocity.

They're connected -- one quality will describe
how the other is changing.

And the way we describe change in mathematics
is through calculus.

Calculus explains how and why things change,
using derivatives,

which help you determine how an equation
is changing, as well as with integrals,

which you can use to calculate the area
under a curve.

Derivatives and integrals themselves are closely
connected. But let's start with derivatives.

You probably won't be able to go straight
from this lesson to your calculus final.

But hopefully, in about 10 minutes, you WILL be
able to understand some of the maths that

scientists have been using to think about
physics, for the last 400 years or so.

And you'll ALSO have a NEW way to fight speeding
tickets. You know, just in case.

Last time, we talked about that unfortunate
incident where you got a speeding ticket.

Your speedometer was broken, but because we
knew your acceleration, we were able to calculate

how fast you were going when the cops pulled
you over.

So now, let's talk about what happens next.

Say the police drive off. You're ready to
get back on the road, so you hit the gas and

But in this scenario, we don't know your acceleration;

we only know how much your position is
changing over time.

In this instance, your position happens
to be equal to the amount of time you've

been driving, squared. So we'd write that
as the equation x = t^2.

20 seconds in, you pass a detector with a
sign that tells you your speed. You keep driving,

foot still on the gas, before you realize
what number you saw on the sign.

And…OH NO! You JUST got a speeding ticket in the last episode, for doing 126 kmh in a 100 kmh zone,

and now the sign says you're going even
FASTER!

Now you want to know if the number on the
detector is accurate -- in other words, you want

to find your velocity, at the exact moment
you passed it.

That velocity is just a measure of your
change in position -- its derivative.

So, to find your velocity, we'll need to
find the derivative of your position.

And in order to determine THAT, we first need
to talk about limits.

Not speed limits -- I mean the derivatives
kind. (pause) I'll explain...

Limits are based on the idea that if you have
a an equation on a graph, you can often predict

what it's going to look like at one point, just by knowing what it looks like at the surrounding points.

For example: let's say you have a graph
of x = t^2 -- from our speeding scenario above

And you want to find out how your position is changing at the exact moment that time is equal to zero.

That's what we'd call the limit as t approaches
zero.

So you take a look at what's happening AROUND
t = 0.

You can probably tell that as we get closer
and closer to t = 0, your value of x is getting

closer to zero, too. That's what mathematicians
mean when they talk about a limit.

Limits are useful because they can help predict
what happens as you make intervals smaller.

An interval is just a range on a graph, it's
the space between two points on the horizontal axis.

So the first thing we can try is calculating
your AVERAGE velocity over the interval from

15 to 20 seconds.
To do that, we use an equation that we talked

about last time --
your average velocity, which is equal to the

change in your position --
divided by the change in time.

Problem is, it's still just an average -- it's
not EXACTLY how fast you were going after

20 seconds of acceleration, when you passed
the detector.

Because of limits, we know that you could
get a little closer to the right number by

calculating your average over smaller and
smaller intervals. Then you'd see that the

number seemed to be getting closer and closer
to 40 meters per second.

Which means that you're going to need to
slow wayyyy down if you don't want to get

But that's the idea of derivatives: you can
use infinitely tiny intervals to figure out

exactly how an equation is changing at any
moment.

You can even come up with an equation to describe
the change. That's exactly what velocity is

-- an equation that describes change in position.

And acceleration describes change in velocity.

So we'd call velocity the derivative of position,
and acceleration the derivative of velocity.

Now, when it comes to how you can express
a derivative in writing, mathematicians have

As the name suggests, it's used for equations
with variables raised to powers, or exponents

For example, x = t^2 would work with the power
rule, because t is raised to the power of 2.

The power rule says that for these kinds of
equations, to calculate the derivative, all

Take the number of that exponent -- in this
case, two -- and stick it in front of the

variable. Then you subtract 1 from the exponent.

So the derivative of x = t^2 is just 2t. Which
means that no matter how many seconds you've

had your foot on the gas, your velocity will
be 2t -- so, double the number of seconds.

After 5 seconds, you were going a modest 10
ms. But after 20 seconds, you were going a

full 40 ms. Which is not good. We'd write
that like this, where dxdt is just a way of

saying that we're taking the derivative of
the part of the equation that involves t.

Or, as a mathematician would put it, we're
taking the derivative of x with respect to t.

You'll also sometimes see this written in
a different way:

If F of T is equal to T squared, then F prime
of T is equal to two T.

Now let's try to find a couple more derivatives
using the power rule.

x = 7t^6 is another power-style equation:
it has a variable, t, raised to a power, 6,

with a number in front of it: 7. The first
thing we do is take the exponent, and stick

But there's already a number in front of t
... 7. So we end up multiplying them: 7 times

6 is 42. Then we subtract 1 from the power
that t is raised to. So we end up with 42t^5.

Same goes for equations where the exponents
are fractions or decimals. So the derivative

of t^½ is half t to the negative one half.