Derivatives: Crash Course Physics #2

Summary

This script explores the language of physics through mathematics, focusing on calculus to describe change. It explains how derivatives, using limits and the power rule, calculate an object's velocity from its position over time. The video also touches on the derivatives of trigonometric functions and e^x, setting the stage for integrals, which will be discussed in a future episode. The script uses a relatable scenario of calculating speed to explain these concepts in a practical way.

Takeaways

• 🔢 The language of physics is mathematics, which is essential for understanding how things relate to each other in the physical world.
• 📉 Physics uses calculus to describe change, with derivatives and integrals being fundamental to this description.
• 🚗 Derivatives are used to determine the rate of change of a quantity, such as velocity being the derivative of position.
• 📚 Limits are a concept in calculus that help predict the behavior of a function at a certain point by examining the function's values around that point.
• 🔍 The Power Rule is a shortcut for finding derivatives of equations with variables raised to a power, simplifying the process of differentiating.
• 📐 Trigonometric functions like sine and cosine are important in physics, and their derivatives are essential for understanding motion and other phenomena.
• 🔄 The derivative of sine is cosine, and vice versa, which is a recurring theme in physics and engineering applications.
• ∞ The derivative of the exponential function e^x is itself e^x, showcasing the unique properties of the exponential function in calculus.
• 🔙 Integrals, the counterpart to derivatives, allow for the calculation of quantities like position from velocity, which will be explored in further lessons.
• 📈 Understanding derivatives and integrals is crucial for predicting and calculating motion and other physical phenomena, with practical applications like fighting speeding tickets.

Q & A

What is the unique language of physics?

The unique language of physics is mathematics, which is used to describe how the world works in a mathematical way.

How are velocity and acceleration related to each other?

Velocity is a measure of change in position, and acceleration is a measure of change in velocity. They are connected in that one quality describes how the other is changing.

What is the purpose of calculus in physics?

Calculus is used in physics to explain how and why things change, using derivatives to determine how an equation is changing and integrals to calculate the area under a curve.

What is the equation that represents the position in the given speeding scenario?

In the speeding scenario, the position is represented by the equation x = t^2, where x is the position and t is the time.

How can you find the velocity at a specific moment using calculus?

To find the velocity at a specific moment, you need to find the derivative of the position function, which represents the rate of change of position with respect to time.

What are limits in calculus and why are they important?

Limits in calculus are based on the idea of predicting the behavior of a function at a point by knowing its behavior at surrounding points. They are important for predicting what happens as intervals become infinitely small.

How is the average velocity calculated over a given interval?

The average velocity over a given interval is calculated by dividing the change in position by the change in time over that interval.

What is the Power Rule in calculus and how is it applied?

The Power Rule in calculus is used for equations with variables raised to powers. It states that to calculate the derivative, you take the exponent, put it in front of the variable, and subtract 1 from the exponent.

What are the derivatives of sin(x) and cos(x)?

The derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x). These derivatives are important in physics for dealing with trigonometric functions.

What is special about the derivative of e^x?

The derivative of e^x is unique because it is e^x itself. This property is one of the defining characteristics of the mathematical constant e.

How can integrals be used in physics in relation to derivatives?

Integrals can be used to reverse the process of derivatives. With integrals, you can use acceleration to find velocity and velocity to find position, effectively working backwards from the rate of change to the original quantities.

Outlines

00:00

📚 The Language of Physics and Calculus

This paragraph introduces the unique language of physics, which is mathematics, and emphasizes the importance of calculus in understanding change. It explains the interconnectedness of position, velocity, and acceleration, and how calculus uses derivatives to measure these changes. The paragraph also outlines the concept of derivatives and integrals, their connection, and the practical application of derivatives in determining velocity from position, using the example of a speeding scenario with the equation x = t^2. Limits are introduced as a fundamental concept in calculus, allowing for the prediction of behavior at a point by examining surrounding points, with an example provided using the graph of x = t^2.

05:05

🔍 Derivatives and the Power Rule

The second paragraph delves into the specifics of calculating derivatives, focusing on the Power Rule for equations with variables raised to an exponent. It explains the process of finding the derivative of x = t^2 and how this can be applied to determine velocity. The paragraph further illustrates the concept with additional examples, including equations with different powers and coefficients. It also touches on the derivatives of trigonometric functions, such as sin(x) and cos(x), and their relationship to each other. The special case of the derivative of e^x is highlighted, showing that it equals e^x itself. The paragraph concludes by emphasizing the utility of these derivative formulas in physics for analyzing motion and preparing for future discussions on integrals.

Keywords

💡Special Language

The term 'special language' refers to the unique terminology and methods of communication that are specific to different scientific disciplines. In the context of the video, it highlights how each field, such as biology, chemistry, and physics, has its way of organizing and conveying knowledge. For instance, biology uses Latin for naming organisms, while chemistry employs a system of prefixes and numerals to describe chemical compositions. This concept is foundational to understanding the unique approach of physics, which relies on mathematics to express its principles.

💡Physics

Physics is a natural science that studies matter, its motion, and behavior through space and time. The script emphasizes that physics communicates its ideas through mathematics, focusing on relationships and changes in the physical world. It is central to the video's theme, as it sets the stage for explaining how mathematical concepts like calculus are essential to understanding and describing physical phenomena.

💡Mathematics

Mathematics is the abstract science of number, quantity, and space, either as abstract concepts or represented by symbols. In the video, it is presented as the 'language of physics,' crucial for describing the world in a precise way. The script uses mathematics to explain concepts such as position, velocity, and acceleration, and how they interrelate, which is vital for comprehending the mechanics of motion.

💡Calculus

Calculus is a branch of mathematics that deals with the study of change and motion, focusing on the concepts of derivatives and integrals. The video explains calculus as the mathematical method used to describe how and why things change. It is integral to the script's narrative, as it is used to illustrate how to calculate rates of change, such as velocity from position, and acceleration from velocity.

💡Derivatives

Derivatives in calculus represent the rate at which a quantity changes with respect to another quantity. The script introduces derivatives as a means to understand how an equation changes, using them to calculate the velocity of an object from its position. For example, the derivative of position with respect to time gives velocity, which is central to the video's explanation of motion.

💡Integrals

Integrals are the counterpart to derivatives in calculus, used to calculate the accumulated change, such as the area under a curve. While the video does not delve deeply into integrals, it mentions them as a way to reverse the process of derivatives, allowing one to find velocity from acceleration and position from velocity, which is crucial for understanding the inverse relationship between these quantities.

💡Limits

Limits are a fundamental concept in calculus that describe the value that a function or sequence 'approaches' as the input approaches some value. The script explains limits as a way to predict the behavior of a function at a specific point by examining the function's values around that point, which is essential for understanding the concept of derivatives.

💡Position

In the context of physics and the video, 'position' refers to the location of an object in space. The script uses the concept of position to introduce the idea of change over time, which is fundamental to the study of motion. It is exemplified by the equation x = t^2, which describes how position changes with time in a specific scenario.

💡Velocity

Velocity is a vector quantity that represents the rate of change of an object's position with respect to time. The video script explains velocity as the derivative of position, which means it is the measure of how fast and in what direction an object is moving. It is used in the narrative to illustrate how calculus can be applied to determine the instantaneous speed of an object.

💡Acceleration

Acceleration is the rate of change of velocity with respect to time. In the script, acceleration is introduced as the derivative of velocity, indicating how quickly the velocity of an object is changing. It is a key concept in the study of motion, as it helps understand how an object's speed increases or decreases over time.

💡Power Rule

The Power Rule is a basic principle in calculus for finding the derivative of a function that involves a variable raised to a power. The video script explains the Power Rule as a shortcut for calculating derivatives of the form x = tn, where 'n' is the exponent. It demonstrates the rule with the example x = t^2, showing how to find the derivative, which is 2t, by using the rule.

💡Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles, particularly right-angled triangles. The script mentions trigonometry in the context of derivatives, explaining how to find the derivatives of sine and cosine functions, which are essential in physics for dealing with periodic phenomena and angular measurements.

💡e^x

The term 'e^x' refers to the exponential function with base 'e', where 'e' is the mathematical constant approximately equal to 2.718. The video script highlights a unique property of 'e^x', stating that its derivative is the function itself. This characteristic is fundamental in various applications of calculus, including natural growth and decay processes.

Highlights

Science disciplines have unique languages for communication, with physics using mathematics.

Physics describes relationships through mathematical concepts like position, velocity, and acceleration.

Calculus is essential in physics for explaining changes through derivatives and integrals.

Derivatives determine how an equation changes, integrals calculate the area under a curve.

The connection between velocity, acceleration, and their derivatives is fundamental in physics.

Derivatives can be used to challenge speeding tickets by calculating exact speed at a moment in time.

Position equation x = t^2 is used to illustrate the concept of finding velocity through derivatives.

Limits are foundational in calculus, predicting behavior at a point by examining surrounding points.

Average velocity can be calculated over intervals but does not provide exact speed at a specific moment.

The concept of limits allows for increasingly precise calculations by shrinking intervals.

Derivatives use infinitely small intervals to determine exact changes in an equation at any moment.

The Power Rule is a shortcut for calculating derivatives of equations with variables raised to powers.

Derivatives of trigonometric functions, such as sin(x) and cos(x), are crucial in physics.

The derivative of sine is cosine, and vice versa, reflecting their periodic nature.

The derivative of e^x is unique, being e^x itself, showcasing the constant's importance in calculus.

Integrals, the counterpart to derivatives, allow for the reverse process in calculus.

This episode of Crash Course Physics introduces limits, derivatives, and their applications in physics.

The episode is produced in association with PBS Digital Studios, offering a variety of educational content.

Transcripts

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.

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 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.

[Theme Music]

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 = 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.

At t = 1, x is 1.

At t = 0.5, x is 0.25.

And at t = 0.1, x is 0.01.

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.

That turns out to be 35 ms.

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

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.

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

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 = 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

you need is one weird trick.

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.

And that's your derivative!

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

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^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.

It works for negative exponents, too -- the derivative of t^-2 is just negative 2 t to

the negative third.

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

Trigonometry -- which we use to calculate the angles and sides of triangles -- is going

to come up a lot in physics, because we'll be using right angle triangles all the time.

So it's a good idea to know how to find the derivatives of sin(x) and cos(x).

Sine tells you that if you have a right angle triangle, and x is an angle in that triangle, then sin(x)

will be the (length of the side opposite, that angle), divided by the (hypotenuse).

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

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

We can actually try to guess the derivative of sin(x) just by looking at its graph.

You can see that the curve has turning points every so often, at x = -90 degrees, x = 90

degrees, and so on -- repeating every 180 degrees.

Meaning, at those points, the equations aren't changing at all -- so the derivative at these

turning points is also going to be exactly zero.

Let's pull up another graph where we'll plot the derivative, and put little dots where

we know it'll be zero.

Now, what's happening between those turning points? Well, from -270 to -90 degrees, sin(x)

is decreasing.

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

Then, from -90 to 90 degrees, sin(x) is increasing -- so it'll have a positive derivative. And so on...

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.

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 … hey, this derivative is looking

a whole lot like the graph of cos(x)!

That’s because it is. The derivative of sine is just cosine, and that is going to

come up a LOT.

So will these, which you can work out on your own by repeating what we just did with the

graphs of sin(x) and cos(x).

Another important derivative that comes up a lot is a very special case, and that’s

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

Yep, that’s it. No matter what.

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 -- about 2.718, but with

more digits after the decimal point that go on forever.

It has all sorts of uses in calculus, but it also shows up when you’re studying things

like finance and probability.

Armed with all these ways to find derivatives, you could take pretty much any equation of

your position and calculate its derivative -- and therefore your velocity.

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

But there's still a whole other part of calculus that we haven't talked about yet -- integrals

-- which will let you do this backwards.

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

But we'll save that for next time.

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

We also talked about a few different kinds of derivatives: powers, constants, trigonometry, and e^x.

Crash Course Physics is produced in association with PBS Digital Studios. You can head over

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

This episode of Crash Course was filmed in the Doctor Cheryl C. Kinney Crash Course Studio

with the help of these amazing people….and our Graphics Team is Thought Cafe.