Derivative definition

Summary

The video script introduces the concept of the derivative of a single-variable function, emphasizing its definition and various notations. It explains the derivative as the limit of the difference quotient as Δx approaches zero, denoted by a prime symbol or Leibniz's dy/dx notation. An example illustrates calculating the instantaneous rate of change, showing how to find the derivative at a specific point and the derivative function. The script concludes by highlighting the importance of understanding different derivative notations for practical applications.

Takeaways

• 📘 The derivative of a function f at a point x_0 is defined as the limit of the difference quotient as Δx approaches 0.
• 🖊 The derivative is denoted by a prime symbol, such as f'(x_0), indicating the rate of change at a specific point.
• 🌱 An example using the derivative is provided with the growth rate of a plant, where the height function is given by h(x) = 2x + 3, and the derivative at 30 days is calculated.
• 📈 The derivative function f'(x) is a generalization that allows for finding the derivative at any point by substituting x_0 with a variable.
• 🔢 The process to find the derivative function involves taking the limit of the difference quotient for a given function, as demonstrated with f(x) = 3x^2.
• 📐 The derivative function for 3x^2 is found to be 6x, showcasing the simplification and cancellation of terms in the limit process.
• 📝 Three notations for the derivative are introduced: prime notation, Leibniz notation (dy/dx), and operator notation (Dx).
• 📋 Prime notation is used to denote both the derivative function and the derivative at a specific point, with the latter indicated by specifying the point.
• 📏 Leibniz notation emphasizes the rate of change, using differentials (dy and dx) instead of differences, and is useful for understanding the derivative as an instantaneous rate of change.
• 🛠 Operator notation uses Dx to denote the derivative, providing a compact way to express the derivative operation.

Q & A

What is the definition of the derivative of a function at a point?

The derivative of a function f at a point x0 is given by the limit as Δx approaches 0 of (f(x0 + Δx) - f(x0)) / Δx.

What does the prime notation represent in calculus?

The prime notation, denoted as f', represents the derivative of a function f at a specific point x0.

How is the instantaneous rate of change related to the derivative?

The instantaneous rate of change at a point x0 is represented by the derivative of the function at that point, which is the same formula used to define the derivative.

What is an example provided in the script to illustrate the concept of a derivative?

An example given is the growth of a plant over time, where the height of the plant is represented by the function h(x) = 2x + 3, and the derivative is used to find the rate of growth after 30 days.

What does the derivative of the function h(x) = 2x + 3 at x = 30 equal to?

The derivative of h(x) at x = 30 is 2, indicating the plant's height is growing at a rate of 2 units per day after one month.

Why is it beneficial to define a derivative function instead of calculating the derivative at a specific point?

Defining a derivative function allows for a more efficient calculation of the derivative at any point of interest by simply plugging in the value of x, rather than recalculating the entire derivative each time.

What is the derivative function of f(x) = 3x^2?

The derivative function of f(x) = 3x^2 is f'(x) = 6x, which can be found by applying the definition of the derivative with x as a variable.

What are the three different notations for the derivative introduced in the script?

The three notations for the derivative are: prime notation (f'), Leibniz notation (dy/dx), and operator notation (D_x).

How does Leibniz notation for the derivative differ from the prime notation?

Leibniz notation uses dy/dx to represent the derivative function, and dy/dx evaluated at x = x0 is denoted by placing a vertical bar (dy/dx)|x=x0 to indicate the derivative at a specific point.

What is the significance of the operator notation for the derivative?

Operator notation, denoted by D_x, signifies the action of taking the derivative of a function and can be used to indicate the derivative evaluated at a point by adding a vertical bar (D_x)|x=x0.

What is the main takeaway from the script regarding derivatives?

The main takeaway is the formal definition of a derivative function and the derivative at a single point, along with an understanding of the three common notations used to represent derivatives.

Outlines

00:00

🌱 Introduction to Derivatives

This paragraph introduces the concept of the derivative of a single variable function. The derivative of a function f at a point x0 is defined using the limit process, which is the instantaneous rate of change. The notation for the derivative includes a prime symbol (f'(x0)) to indicate the derivative at a specific point. An example is given where the height of a plant over time is modeled by a function, and the derivative is used to find the rate of growth after one month. The process involves calculating the limit as Δx approaches 0 of the difference in function values divided by Δx. The example concludes with finding the derivative at x0 = 30 days, resulting in a growth rate of 2 units per day. The paragraph also touches on the idea of deriving a general derivative function that can be evaluated at any point by substituting x0 with the variable of interest.

05:02

📚 Derivative Notations and Calculations

This paragraph delves into different notations used to represent derivatives and demonstrates how to calculate the derivative of a function. The derivative function f'(x) is defined using the limit process, and an example is provided to calculate the derivative of f(x) = 3x^2. The calculation involves expanding the function at x + Δx, simplifying, and then taking the limit as Δx approaches 0. The result is the derivative function f'(x) = 6x. The paragraph introduces three notations for derivatives: prime notation (f'(x)), Leibniz notation (dy/dx), and operator notation (D_x). Each notation is explained, and the context in which they are used is discussed. The paragraph concludes with a summary of the key points, emphasizing the formal definition of the derivative function and the understanding of various notations.

Keywords

💡Derivative

The derivative of a function represents the rate at which the function's output changes with respect to its input. In the video, the derivative is formally defined as the limit of the difference quotient as the change in the input approaches zero. It is central to the theme of the video, which is to understand how fast a quantity is changing at any given point, exemplified by the growth rate of a plant over time.

💡Single Variable Function

A single variable function is a mathematical function that relates each value of a single input variable to one output value. The video discusses the derivative of such functions, emphasizing how to calculate the rate of change for these functions, which is crucial for understanding the behavior of various phenomena, like the growth of a plant.

💡Notation

Notation in mathematics refers to the symbols and abbreviations used to denote mathematical concepts. The video explores different notations for derivatives, such as prime notation (f'(x)), Leibniz notation (dy/dx), and operator notation (Dx f(x)). Understanding these notations is essential for communicating mathematical ideas clearly and effectively.

💡Instantaneous Rate of Change

The instantaneous rate of change is the rate at which a quantity changes at a specific instant. In the context of the video, it is shown to be equivalent to the derivative of a function at a point. The example of a plant's growth rate illustrates how the instantaneous rate of change can be calculated using the derivative.

💡Limit

In calculus, a limit is the value that a function or sequence approaches as the input (or index) approaches some value. The video defines the derivative using limits, specifically as the limit of the difference quotient as the change in the input approaches zero. This concept is fundamental to understanding derivatives and their calculation.

💡Delta X (Δx)

Delta X, often denoted as Δx, represents a small change in the input variable of a function. In the video, Δx is used in the difference quotient to calculate the derivative, highlighting its role in finding the rate of change as the input approaches a certain value.

💡F Prime (f')

F Prime, denoted as f', is a notation used to represent the derivative of a function f. The video explains how f'(x) at a certain point x can be calculated using the limit process, and how it can be used to find the rate of change at that point, such as the daily growth rate of a plant after one month.

💡Leibniz Notation

Leibniz notation, introduced by Gottfried Leibniz, is a way to denote the derivative of a function, written as dy/dx. The video mentions this notation as a historical and practical method to express derivatives, emphasizing its connection to the concept of differentials and the average rate of change.

💡Operator Notation

Operator notation uses the symbol D to denote the derivative operation. In the video, it is presented as an alternative way to express derivatives, such as Dx f(x), which indicates the derivative of the function f with respect to x. This notation is particularly useful for indicating the operation of differentiation.

💡Variable

A variable in mathematics is a symbol that represents a value that can change. The video discusses how to define a derivative function where the input x is a variable, allowing for the calculation of the derivative at any point of interest. This concept is crucial for generalizing the derivative calculation beyond a single point.

💡Derivative Function

A derivative function is a function that gives the derivative of another function at every point in its domain. The video explains how to derive such a function from a given function, like finding the derivative function for f(x) = 3x^2, which results in f'(x) = 6x. This is a powerful tool for analyzing how quantities change across a range of values.

Highlights

Definition of the derivative of a single variable function f is introduced.

Derivative formula presented: limit as Δx approaches 0 of [f(x₀ + Δx) - f(x₀)] / Δx.

Prime notation (f') explained for indicating the derivative of a function.

Instantaneous rate of change at a point x₀ is discussed in relation to the derivative.

Example of calculating the derivative to determine the growth rate of a plant over time.

Strategy to calculate the derivative at 30 days using the definition of the derivative.

Simplification of the derivative formula resulting in a growth rate of 2 units per day for the plant.

Introduction to the concept of a derivative function, f'(x), as a variable.

Derivative function defined as f'(x) = limit as Δx approaches 0 of [f(x + Δx) - f(x)] / Δx.

Explanation of how to find the derivative at another point by using the derivative function.

Derivative function for f(x) = 3x² is calculated to be 6x.

Different notations for the derivative are introduced: prime, Leibniz, and operator notations.

Leibniz notation (dy/dx) explained for the derivative function and at a specific point.

Operator notation (D_x) described for denoting the derivative of a function.

Takeaways include the formal definition of the derivative and understanding of different notations.

Transcripts

hi again so our goals for today are to

First formally Define the derivative of

a single variable function f

and two to explore different notations

for the derivative

let's go right ahead and start with the

definition

the derivative of a function f at a

point x 0 is given by the following

formula

limit as Delta X approaches 0 of f of x

naught plus Delta x minus f of x naught

divided by Delta X

in terms of notation we use a prime sign

so this little apostrophe to indicate a

derivative so F Prime

at X zero

is indicated as such

note that this is exactly the same

formula that we saw for the

instantaneous rate of change

at the point x zero

let's look at an example

so you bought a plant to decorate your

room and you want to know how fast your

plant will be growing

you found online that the height of this

plant is given by the following function

age of X is equal to 2x plus 3 where X

is the number of days after repotting

so here's your plant and age is the

function that tells us its height over

time

so we want to find the instantaneous

rate of change after one month

so X naught is going to be equal to 30.

so 30 days after you bite

our strategy will be to use the

definition of the derivative that we

have above

we want to calculate H Prime the

derivative

at 30 days

that's going to be equal to the Limit as

Delta X approaches zero of the function

evaluated at 30 plus Delta x minus the

function evaluated at 30 days

and we divide that by Delta X we plug in

30 plus Delta X to our function and we

get 2 times 30 plus Delta X plus 3 and

then minus 2 times 30 and minus 3. so

that's the function value at 30.

all of that divided by Delta X

so if we simplify this out the threes

cancel out the minus 60 cancels out with

a positive 60.

but we are left with two times Delta X

over here

and then it's all divided by Delta X

so the Delta X's cancel out and we have

limit as Delta X approaches 0 of 2.

which is just two there's no Delta X in

the formula to be plugged in so that's

telling us that the height of our plant

after one month is growing at a rate of

2 units per day

a quick note the method presented

previously allows us to calculate the

derivative at a specific point x0

if we wanted to find the derivative or

the instantaneous rate of change at

another point we would need to redo the

entire calculation

more efficiently we may use x0 as a

variable

and thus find a derivative function that

is you pick an X naught and then I'll

plug it into this function and tell you

the derivative at that point

that way we just need to find one

function and plug in X naught whenever

we're interested so let's define this

derivative function we'll say that the

derivative function f Prime of X is

given by

F Prime of X is equal to the Limit as

Delta X approaches 0 of f of x plus

Delta x minus f of x divided by Delta X

as we said before X here is a variable

we can plug in and replace it by our

values of Interest

We compare that with the previous

definition that we had F Prime at X

naught where X naught was a fixed value

the formulas look the same it's just the

interpretation that changes so we go

from having fixed values to having

variables

see this distinction in an example

say we want to find the derivative

function of f of x equals 3x squared so

we're using this new definition from the

left where X is a variable then we'll

have

F Prime of X is equal to the Limit as

Delta X approaches zero of f of x plus

Delta x minus f of x divided by Delta X

the steps are very similar we're going

to plug in X Plus Delta X into our

function of Interest

that would give us 3 times X Plus Delta

X quantity squared and we subtract off

the original function minus 3x squared

all of that divided by Delta X

we then have the limit as Delta X

approaches zero of let's expand out this

term we get three times x squared plus

two x Delta X Plus Delta x squared minus

the original function 3x squared all

divided by Delta X

then we simplify out multiply the three

you have the limit as Delta X approaches

0 of 3x squared plus six x Delta X plus

three Delta x squared minus three x

squared all divided by Delta X

we see here that the 3x squared cancels

and we're left with 6X Delta X plus

three Delta x squared all over Delta X

and still the limit as Delta X

approaches zero of that quantity

as we can see there is a Delta X in all

of the remaining terms we can cancel one

out

after we do that we can calculate the

limit by taking Delta X to Zero you see

that there's only one term left with a

Delta X and that term goes to zero so

all we're left with is six x

that is the derivative function of 3x

squared is equal to 6x

so if you want to find it at a specific

point now we just replace X by that

point of Interest this is a very

versatile it can help us out in many

situations

it's important to see some different

notations and see how the derivative can

be represented so let's go ahead and

introduce those notations so we've seen

the prime notation that is if I'm given

a function f of x then we denote its

derivative as F Prime of x

and that's the derivative function but

we can also indicate the derivative at a

point by saying F Prime of X naught so

that's the derivative at the point x

equals x naught the second notation

comes from mathematician Gottfried

leibniz so it's called the leibniz

notation and

in the scenario where we're given a

function f of x we denote the derivative

by d y d x that's the derivative

function

if we wanted to denote the derivative at

a point we use the notation d y d x

and we put a vertical bar

indicating that it has to be evaluated

at x equals x naught

so this is the derivative at a given

point

this notation reminds us that this is

coming from an average rate of change an

average rate of change was given by

Delta y divided by Delta X as we take

the limits of these quantities tending

to zero

the Deltas tend to D in this notation so

instead of Deltas we have the

differentials d y and DX the final

notation we'll present today is called

the operator notation and again if we're

given a function y equals f of x we

denote the derivative by a capital D

with a subscript X

that's telling us that we're doing

something to the function f we're taking

its derivative

we can similarly indicate that it's

being evaluated at a point using a

vertical bar

so some takeaways from this video

we have formally defined the concept of

a derivative function but also of the

derivative at a single point

and we looked at three different types

of common notations for the derivative

that's all for now