Derivative as a concept | Derivatives introduction | AP Calculus AB | Khan Academy

Summary

This instructional video delves into the concept of slope as a measure of a line's rate of change. It introduces the idea of calculating slope using the change in y over the change in x, or 'rise over run.' The video then transitions into the realm of calculus, exploring the instantaneous rate of change along a curve, which is not constant like a line's slope. The focus is on finding the derivative, represented as dy/dx or f'(x), which signifies the slope of the tangent line at a specific point. This derivative is central to understanding differential calculus and is approached through the concept of limits as the change in x approaches zero.

Takeaways

• 📏 The concept of slope represents the rate of change of a vertical variable with respect to a horizontal variable.
• 📈 Calculus extends the idea of slope to include the instantaneous rate of change, even for curves where the rate of change is not constant.
• 🔍 To find the slope of a line, one can select two points and calculate the change in y over the change in x, often referred to as 'rise over run'.
• 📉 For a curve, the average rate of change can be found by calculating the slope of the secant line connecting two points on the curve.
• 🎯 The instantaneous rate of change at a specific point on a curve is represented by the slope of the tangent line at that point.
• 🏃‍♂️ The concept of instantaneous rate of change is exemplified by calculating the speed of a sprinter like Usain Bolt at a particular instant.
• 📐 Leibniz's notation, dy/dx, is used to denote the slope of the tangent line, which is the derivative and represents the instantaneous rate of change.
• 🔢 The derivative can also be represented using Lagrange's notation, f'(x), where f is a function and f' denotes its derivative at a given x.
• 🧮 Calculating derivatives involves taking the limit of the ratio of the change in y to the change in x as the change in x approaches zero.
• 🔮 Future lessons in calculus will provide tools to calculate derivatives for any given point and develop general equations for derivatives.

Q & A

What is the slope of a line?

The slope of a line is a measure of its steepness and indicates the rate of change of a vertical variable with respect to a horizontal variable. It is calculated as the change in y (the rise) over the change in x (the run), often described as 'rise over run'.

Why is the slope constant for any line?

The slope is constant for any line because it represents a consistent rate of change. No matter which two points on the line are chosen, the calculated slope remains the same, reflecting the linear and uniform nature of the line.

What is the difference between average rate of change and instantaneous rate of change?

The average rate of change is calculated over a segment of a curve or line, representing the slope of the secant line connecting two points. In contrast, the instantaneous rate of change is the rate of change at a specific point, which can be found by calculating the slope of the tangent line at that point.

How is the concept of a tangent line used in calculus?

In calculus, a tangent line is used to determine the instantaneous rate of change at a specific point on a curve. The slope of the tangent line at that point represents the derivative, which is the instantaneous rate of change.

What is the significance of the derivative in differential calculus?

The derivative in differential calculus is significant because it represents the slope of the tangent line to a curve at a given point, which is the instantaneous rate of change. This concept is central to understanding how quantities change at any given moment.

Who are the fathers of calculus, and what is their contribution to the notation of derivatives?

Isaac Newton and Gottfried Wilhelm Leibniz are considered the fathers of calculus. Leibniz contributed to the notation of derivatives with his differential notation, denoting the derivative as dy/dx, which signifies an infinitesimally small change in y over an infinitesimally small change in x.

What is Leibniz's notation for the derivative?

Leibniz's notation for the derivative is dy/dx, which represents the ratio of an infinitesimally small change in y to an infinitesimally small change in x, especially as the change in x approaches zero.

What is another notation for the derivative besides Leibniz's?

Another notation for the derivative is Lagrange's notation, where the derivative of a function y = f(x) is denoted as f'(x), indicating the slope of the tangent line at a given x-value.

How does the concept of limits relate to finding the derivative?

The concept of limits is fundamental to finding the derivative because it involves taking the limit of the ratio of the change in y to the change in x as the change in x approaches zero, which mathematically defines the derivative.

What is the physical interpretation of the derivative in the context of motion?

In the context of motion, the derivative represents the instantaneous velocity of an object at a specific moment in time. If y represents position and x represents time, then the derivative dy/dx gives the speed of the object at any given instant.

What are some other notations for the derivative that might be seen in physics or math classes?

In physics or math classes, one might see the derivative notated as y with a dot over it (ẏ), which denotes the rate of change of y with respect to time. Another common notation is y', which is often used in mathematical contexts to represent the derivative of y.

Outlines

00:00

📏 Understanding Slope and the Concept of Derivatives

The paragraph introduces the concept of slope as a measure of the rate of change between two variables, typically represented on a Cartesian plane with a vertical y-axis and a horizontal x-axis. The instructor explains how to calculate the slope by selecting two points on a line and determining the change in y over the change in x, often referred to as 'rise over run.' The paragraph then transitions into the realm of calculus, where the focus is on understanding not just the average rate of change but the instantaneous rate of change at a specific point on a curve. The idea of a tangent line, which touches the curve at a single point and represents the instantaneous rate of change at that point, is introduced. The concept of a derivative, represented as the slope of the tangent line, is highlighted as a foundational element of differential calculus, with an emphasis on its importance in understanding the rate of change at any given moment.

05:00

📘 Calculating Derivatives: Notations and Methods

This paragraph delves into the different notations used to represent derivatives, starting with Leibniz's notation, which uses 'dy/dx' to denote the derivative, reflecting the idea of a slope as a ratio of infinitesimal changes in y to x. The concept of approaching the derivative by considering the limit as the change in x approaches zero is discussed, which is fundamental to the differential calculus approach. Other notations, such as 'f' prime of x (f'(x)) to denote the derivative of a function 'f(x)', are introduced, along with the less common 'y with a dot over it' (ẏ), which is often used in physics to represent the derivative. The paragraph concludes by emphasizing the upcoming development of tools within calculus to calculate these derivatives, hinting at the use of limits and the anticipation of deriving general equations for any given point.

Keywords

💡Slope

Slope is a fundamental concept in mathematics, particularly in the context of lines and linear equations. It represents the steepness or incline of a line and is defined as the ratio of the vertical change (change in y) to the horizontal change (change in x) between two points on the line. In the video, the instructor uses the slope to introduce the idea of rate of change, which is essential for understanding calculus. The example given is calculating the slope by selecting two points on a line and determining the change in y over the change in x, illustrating the concept with a visual aid of a line on the coordinate plane.

💡Rate of Change

Rate of change is a central theme in the video and is closely related to the concept of slope. It describes how one quantity changes in relation to another quantity. The video explains that slope is a measure of the rate of change of a vertical variable (like y) with respect to a horizontal variable (like x). The instructor emphasizes that while slope is a constant rate of change for a line, calculus extends this idea to include the instantaneous rate of change for curves, which can be variable.

💡Instantaneous Rate of Change

Instantaneous rate of change is a key concept in calculus that the video aims to explain. It refers to the rate at which a quantity is changing at a specific instant or at a particular point. The video uses the analogy of a tangent line to a curve at a specific point to represent the instantaneous rate of change. This is contrasted with the average rate of change, which is calculated over a distance, and the video suggests that calculus tools will allow for the calculation of these rates even when they are not constant.

💡Derivative

The derivative is a core concept in calculus and is introduced in the video as the mathematical representation of the instantaneous rate of change. It is denoted as the slope of the tangent line at a specific point on a curve. The video explains that the derivative is a way to quantify how quickly a function is changing at any given moment, which is crucial for understanding the behavior of functions and their rates of change.

💡Leibniz's Notation

Leibniz's notation is one of the ways to represent derivatives, named after the mathematician Gottfried Wilhelm Leibniz, who is considered one of the founders of calculus. In the video, the instructor explains that this notation uses 'dy/dx' to denote the derivative, which signifies the infinitesimal change in y for an infinitesimal change in x. This notation is particularly useful for understanding the concept of the limit as the change in x approaches zero.

💡Secant Line

A secant line is a line that intersects a curve at two points and is used to approximate the average rate of change between those points. The video uses the concept of secant lines to illustrate how the average rate of change can vary depending on the points chosen on the curve. It contrasts with the tangent line, which touches the curve at exactly one point and represents the instantaneous rate of change at that point.

💡Tangent Line

A tangent line is a line that touches a curve at exactly one point, and its slope represents the instantaneous rate of change at that point. The video explains that the tangent line is crucial for understanding the concept of the derivative. It is used to visualize and calculate the rate at which a function is changing at a specific instant, which is a fundamental operation in differential calculus.

💡Limit

The concept of a limit is fundamental to calculus and is mentioned in the video as a tool for calculating derivatives. A limit is a value that a function or a sequence approaches as the input (or index) approaches a certain value. In the context of derivatives, the limit is used to find the derivative by considering the ratio of the change in y to the change in x as the change in x approaches zero.

💡Lagrange Notation

Lagrange notation, also known as the prime notation, is another way to denote derivatives. If a function is represented as y = f(x), then its derivative at a point x is denoted as f'(x). The video briefly mentions this notation as an alternative to Leibniz's notation, indicating that it is less common in calculus classes but might be seen in physics.

💡Differential Notation

Differential notation is a term used in the video to describe Leibniz's approach to calculus, which involves considering very small changes (infinitesimal differences) in variables. This notation uses 'd' to represent these infinitesimals, as in 'dy' for an infinitesimally small change in y and 'dx' for an infinitesimally small change in x. It is a way to express the concept of a derivative, focusing on the limit of the ratio of these infinitesimals as dx approaches zero.

Highlights

Introduction to the concept of slope as the rate of change of a vertical variable with respect to a horizontal variable.

Explanation of calculating slope by picking two points and finding the change in x and y.

Definition of slope as change in y over change in x, also known as 'rise over run'.

Description of a line having a constant rate of change, hence a constant slope.

Introduction to calculus as a tool to analyze the rate of change beyond just lines.

Discussion on the concept of instantaneous rate of change for curves.

Example of calculating average rate of change between two points on a curve.

Illustration of how the average rate of change varies with different points on a curve.

Introduction to the concept of a tangent line and its relevance to instantaneous rate of change.

Explanation of the tangent line as the line that touches the curve at a single point.

Derivative as the slope of the tangent line and its significance in differential calculus.

Leibniz's notation for denoting the derivative as dy/dx, representing infinitesimally small changes.

Differentiation between Leibniz's notation and the concept of secant lines.

Lagrange notation for derivatives, f'(x), indicating the slope of the tangent line at a point.

Alternative notations for derivatives, such as y with a dot over it or y', used in physics and math classes.

Anticipation of building tools to calculate derivatives and their general equations.

The importance of limits in understanding derivatives as changes in x approach zero.

Transcripts

- [Instructor] You are likely already familiar with the idea

of a slope of a line.

If you're not, I encourage you to review it on Khan Academy,

but all it is, it's describing the rate of change

of a vertical variable

with respect to a horizontal variable,

so for example, here I have our classic y axis

in the vertical direction and x axis

in the horizontal direction,

and if I wanted to figure out the slope of this line,

I could pick two points,

say that point and that point.

I could say, "Okay, from this point to this point,

what is my change in x?"

Well, my change in x would be this distance right over here,

change in x,

the Greek letter delta, this triangle here.

It's just shorthand for "change," so change in x,

and I could also calculate the change in y,

so this point going up to that point, our change in y,

would be this, right over here, our change in y,

and then, we would define slope, or we have defined slope

as change in y over change in x,

so slope is equal to the rate of change

of our vertical variable

over the rate of change of our horizontal variable,

sometimes described as rise over run,

and for any line, it's associated with a slope

because it has a constant rate of change.

If you took any two points on this line,

no matter how far apart or no matter how close together,

anywhere they sit on the line,

if you were to do this calculation,

you would get the same slope.

That's what makes it a line,

but what's fascinating

about calculus is we're going to build the tools

so that we can think about the rate of change not just

of a line, which we've called "slope" in the past,

we can think about the rate of change,

the instantaneous rate of change of a curve,

of something whose rate

of change is possibly constantly changing.

So for example, here's a curve where the rate of change of y

with respect to x is constantly changing,

even if we wanted to use our traditional tools.

If we said, "Okay, we can calculate the average rate

of change," let's say between this point and this point.

Well, what would it be?

Well, the average rate of change between this point and

this point would be the slope of the line

that connects them,

so it would be the slope of this line of the secant line,

but if we picked two different points,

we pick this point and this point,

the average rate of change

between those points all of a sudden looks quite different.

It looks like it has a higher slope.

So even when we take the slopes between two points

on the line, the secant lines,

you can see that those slopes are changing,

but what if we wanted to ask ourselves

an even more interesting question.

What is the instantaneous rate of change at a point?

So for example, how fast is y changing

with respect to x exactly at that point,

exactly when x is equal to that value.

Let's call it x one.

Well, one way you could think about it is

what if we could draw a tangent line to this point,

a line that just touches the graph right over there,

and we can calculate the slope of that line?

Well, that should be the rate of change at that point,

the instantaneous rate of change.

So in this case,

the tangent line might look something like that.

If we know the slope of this,

well then we could say that

that's the instantaneous rate of change at that point.

Why do I say instantaneous rate of change?

Well, think about the video on these sprinters,

Usain Bolt example.

If we wanted to figure out the speed of Usain Bolt

at a given instant, well maybe this describes his position

with respect to time if y was position and x is time.

Usually, you would see t as time, but let's say x is time,

so then, if were talking about right at this time,

we're talking about the instantaneous rate,

and this idea is the central idea of differential calculus,

and it's known as a derivative,

the slope of the tangent line, which you could also view

as the instantaneous rate of change.

I'm putting an exclamation mark

because it's so conceptually important here.

So how can we denote a derivative?

One way is known as Leibniz's notation,

and Leibniz is one of the fathers of calculus

along with Isaac Newton,

and his notation, you would denote the slope

of the tangent line

as equaling dy over dx.

Now why do I like this notation?

Because it really comes from this idea of a slope,

which is change in y over change in x.

As you'll see in future videos,

one way to think about the slope

of the tangent line is, well,

let's calculate the slope of secant lines.

Let's say between that point and that point,

but then let's get even closer,

say that point and that point,

and then let's get even closer

and that point and that point,

and then let's get even closer,

and let's see what happens as the change

in x approaches zero,

and so using these d's instead of deltas,

this was Leibniz's way of saying,

"Hey, what happens if my changes

in, say, x become close to zero?"

So this idea,

this is known as sometimes differential notation,

Leibniz's notation, is instead of just change

in y over change in x, super small changes in y

for a super small change in x,

especially as the change in x approaches zero,

and as you will see,

that is how we will calculate the derivative.

Now, there's other notations.

If this curve is described as y is equal to f of x.

The slope of the tangent line

at that point could be denoted

as equaling f prime of x one.

So this notation takes a little bit of time getting used to,

the Lagrange notation.

It's saying f prime is representing the derivative.

It's telling us the slope of the tangent line

for a given point,

so if you input an x into this function into f,

you're getting the corresponding y value.

If you input an x into f prime,

you're getting the slope of the tangent line at that point.

Now, another notation that you'll see less likely

in a calculus class but you might see in a physics class

is the notation y with a dot over it,

so you could write this is y with a dot over it,

which also denotes the derivative.

You might also see y prime.

This would be more common in a math class.

Now as we march forward in our calculus adventure,

we will build the tools to actually calculate these things,

and if you're already familiar with limits,

they will be very useful, as you could imagine,

'cause we're really going to be taking the limit

of our change in y over change in x as our change

in x approaches zero,

and we're not just going to be able to figure it out

for a point.

We're going to be able to figure out general equations

that described the derivative for any given point,

so be very, very excited.