Shadowing-Übung: Derivatives: Crash Course Physics #2 - Englisch Sprechen Lernen mit Video

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.

Physics has to communicate its ideas differently.

The language of physics is mathematics.

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 to 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 zoom forward, moving faster and faster.

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 equals t squared.

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 km an hour in a 100 km an hour zone.

And now the detector 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 changing 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.

I'll explain.

Limits are based on the idea that if you have 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 equals t squared 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 is what we call the limit as t approaches zero.

So you take a look around at what's happening around t equals zero.

At t equals one, x is one.

At t equals zero point five, x is zero point two five.

And at t equals 0.1, x is 0.01.

You can probably tell that as we get closer and closer to t equals zero, 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.

That turns out to be 35 meters per second.

Problem is, it's 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 seems to be getting closer and closer to 40 meters per second.

Which means you're going to need to slow way down

if you don't want to get your second speeding ticket of the day.

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.

An acceleration describes change in velocity.

So we 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 come up with shortcuts.

Like what's known as the power rule.

As the name suggests, it's used for equations with variables raised to powers or exponents, as long as the exponent is a number.

For example, x equals t squared 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 you need is one weird trick.

Take the number of that exponent, in this case 2, and stick it in front of the variable.

Then you subtract 1 from the exponent.

And that's your derivative!

So the derivative of x equals t squared is just 2t, Which means that no matter how many seconds you've got 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 meters per second.

But after 20 seconds, you were going a full 40 meters per second, which is not good.

We'd write that like this, where dx over dt 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 is equal to t squared, then f prime of t is equal to 2t.

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

X equals 7t to the power 6 is another power-style equation.

It has a variable t raised to the power 6, with a number in front of it, 7.

The first thing we do is take the exponent and stick it in front of the variable.

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 to the power 5.

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

So the derivative of t to the power half is half t to the negative one-half.

It works for negative exponents, too.

The derivative of t to the power minus 2 is just negative 2t to the negative third.

Now, there are a few more equations whose derivatives you should understand.

100

Trigonometry, which we use to calculate the angles and size of triangles, is going to This is going to come up a lot in physics, because we'll be using right-angled triangles all the time.

101

So it's a good idea to know how to find the derivatives of sine x and cosine x.

102

Sine tells you that if you have a right-angled triangle, and x is the angle in that triangle, then sine x will be the length of the side opposite that angle, divided by the hypotenuse.

103

Cosine does the same thing, just with the side next to the angle, divided by the hypotenuse.

104

So their graphs will tell you what those ratios will be, depending on the angle.

105

We can actually try to guess the derivative of sine x, just by looking at its graph.

106

We can see that the curve has turning points every so often, at x equals minus 90 degrees, x equals 90 degrees, and so on, repeating every 180 degrees.

107

Meaning at those points, the equations aren't changing at all.

108

So the derivative at these turning points is also going to be exactly zero.

109

Let's pull up another graph where we'll plot the derivative and put little dots where we'll know it'll be zero.

110

Now, what's happening between those turning points?

111

Well, from minus 270 to minus 90 degrees, sine x is decreasing.

112

In other words, its change, and therefore its derivative, must be negative.

113

Then, from minus 90 to 90 degrees, sine x is increasing, so it'll have a positive derivative, and so on.

114

There are actually a lot more clues in this graph to help us find the derivative, but we already know enough to make a decent guess.

115

If we smoothly connect the dots on the graph of our derivative, keeping in mind where the curve should be positive and where it should be negative,

116

hey, this derivative is looking a whole lot like the graph of cosine x.

117

That's because it is!

118

The derivative of sine is just cosine, and that's going to come up a lot.

119

And so will these, which you can work out on your own by repeating what we just did with the graphs of sine x

120

and cosine x.

121

Another important derivative that comes up a lot is a very special case, and that's e to the x.

122

The derivative of e x is just...e x.

123

Yep, and that's it, no matter what.

124

In fact, that's one way to define e, which is kind of like pi, in the sense that it's a simple letter, representing a very specific, irrational number,

125

about 2.718, with more digits after the decimal point that go on forever.

126

It has all sorts of uses in calculus, but it also shows up when you're studying things like finance and probability.

127

Armed with all these ways to find derivatives, you could pretty much take any equation of your position and calculate its derivative, and therefore, your velocity.

128

In the same way, you could take the derivative of your velocity and find your acceleration.

129

But there's a whole other part of calculus that we haven't even talked about yet.

130

Integrals, which will let you do this backwards.

131

With integrals, you can use your acceleration to find your velocity, and your velocity to find your position.

132

But we'll save that for next time.

133

Today you learned about limits and that derivatives use them to describe how an equation is changing.

134

We also talked about a few different kinds of derivatives – powers, constants, trigonometry, and e to the power x.

135

Crash Course Physics is produced in association with PBS Digital Studios.

136

You can head over to their channel to check out amazing shows like Deep Look, The Good Stuff, and PBS Space Time.

137

This episode of Crash Course is filmed in the Dr. Cheryl C.

138

Kinney Crash Course Studio, the help of these amazing people and our graphics team is Thought Cafe.

📺 Gleicher Kanal

Vectors and 2D Motion: Crash Course Physics #4

TOEFL PrepCambridge Exam PrepUniversity Students

How Often do Asteroids Hit the Earth?: Crash Course Geology #20

IELTS PrepTOEFL PrepUniversity Students

What's all the Yellen About? Monetary Policy and the Federal Reserve: Crash Course Economics #10

IELTS PrepBusiness EnglishUniversity Students

Defense Against the Dark Arts of Influence: Crash Course Business Soft Skills #2

Business EnglishUniversity StudentsWorking Adults

Intro to History of Science: Crash Course History of Science #1

IELTS PrepTOEFL PrepUniversity Students

✨ Empfohlenes Video

549 Sätze

Philosophy of Science Lecture #1: Introduction

IELTS PrepTOEFL PrepCambridge Exam Prep

181 Sätze

The Limit (do not use L'Hospital rule)

Cambridge Exam PrepUniversity StudentsWorking Adults

221 Sätze

Derivative as slope of a tangent line | Taking derivatives | Differential Calculus | Khan Academy

Cambridge Exam PrepUniversity StudentsWorking Adults

190 Sätze

Rates and Returns – Module 1 – Quantitative Methods – CFA® Level I 2026

Cambridge Exam PrepUniversity StudentsWorking Adults

114 Sätze

Newton, Leibniz, and Usain Bolt | Derivatives introduction | AP Calculus AB | Khan Academy

Cambridge Exam PrepUniversity StudentsWorking Adults

55 Sätze

Convergence and Divergence: The Return of Sequences and Series

Cambridge Exam PrepUniversity StudentsWorking Adults

148 Sätze

First Law of Thermodynamics, Basic Introduction - Internal Energy, Heat and Work - Chemistry

IELTS PrepTOEFL PrepCambridge Exam Prep

156 Sätze

How Quantum Batteries Break Every Rule of Charging

Business EnglishUniversity StudentsWorking Adults

119 Sätze

Mathematics and Physics | Lesson 2 | Understanding the Connection

University StudentsWorking Adults

102 Sätze

Orbital Mechanics - Newton's Laws

TOEFL PrepCambridge Exam PrepUniversity Students

Was ist die Shadowing-Technik?

Shadowing ist eine wissenschaftlich fundierte Sprachlerntechnik, die ursprünglich für die professionelle Dolmetscherausbildung entwickelt und durch den Polyglotten Dr. Alexander Arguelles populär gemacht wurde. Die Methode ist einfach aber wirkungsvoll: Du hörst englisches Audio von Muttersprachlern und wiederholst es sofort laut — wie ein Schatten, der dem Sprecher mit nur 1–2 Sekunden Verzögerung folgt. Anders als passives Hören oder Grammatikübungen zwingt Shadowing dein Gehirn und deine Mundmuskulatur, gleichzeitig echte Sprachmuster zu verarbeiten und zu reproduzieren. Studien zeigen, dass es Aussprachegenauigkeit, Intonation, Rhythmus, verbundene Sprache, Hörverständnis und Sprechflüssigkeit signifikant verbessert — was es zu einer der effektivsten Methoden für die IELTS Speaking-Vorbereitung und reale englische Kommunikation macht.

Shadowing-Technik: die vollständige Schritt-für-Schritt-Anleitung lesen →

Shadowing-Übung: Derivatives: Crash Course Physics #2 - Englisch Sprechen Lernen mit Video

Vectors and 2D Motion: Crash Course Physics #4

How Often do Asteroids Hit the Earth?: Crash Course Geology #20

What's all the Yellen About? Monetary Policy and the Federal Reserve: Crash Course Economics #10

Unit Conversion & Significant Figures: Crash Course Chemistry #2

What Is Myth? Crash Course World Mythology #1

Mesopotamia: Crash Course World History #3

Crash Course Sex Ed Preview

The Agricultural Revolution: Crash Course World History #1

Indus Valley Civilization: Crash Course World History #2

Ancient Egypt: Crash Course World History #4

What is Engineering?: Crash Course Engineering #1

The Persians & Greeks: Crash Course World History #5

Supervised Learning: Crash Course AI #2

The Periodic Table: Crash Course Chemistry #4

Where Do Diamonds Come From?: Crash Course Geology #8

Why You Need Trust to Do Business: Crash Course Business - Soft Skills #1

Where Did Democracy Come From, and Does It Still Work?

The Nucleus: Crash Course Chemistry #1

Defense Against the Dark Arts of Influence: Crash Course Business Soft Skills #2

Intro to History of Science: Crash Course History of Science #1

Philosophy of Science Lecture #1: Introduction

The Limit (do not use L'Hospital rule)

Derivative as slope of a tangent line | Taking derivatives | Differential Calculus | Khan Academy

Rates and Returns – Module 1 – Quantitative Methods – CFA® Level I 2026

Newton, Leibniz, and Usain Bolt | Derivatives introduction | AP Calculus AB | Khan Academy

Convergence and Divergence: The Return of Sequences and Series

First Law of Thermodynamics, Basic Introduction - Internal Energy, Heat and Work - Chemistry

How Quantum Batteries Break Every Rule of Charging

Mathematics and Physics | Lesson 2 | Understanding the Connection

Orbital Mechanics - Newton's Laws

Kontext und Hintergrund

Top 5 Phrasen für die tägliche Kommunikation

Schritt-für-Schritt-Shadowing-Anleitung

Was ist die Shadowing-Technik?