Derivative as a concept | Derivatives introduction | AP Calculus AB | Khan Academy
Summary
TLDRThis 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
📏 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.
📘 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.
Mindmap
Keywords
💡Slope
💡Rate of Change
💡Instantaneous Rate of Change
💡Derivative
💡Leibniz's Notation
💡Secant Line
💡Tangent Line
💡Limit
💡Lagrange Notation
💡Differential Notation
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.
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