Calculus- Lesson 8 | Derivative of a Function | Don't Memorise

Summary

This script explores the concept of derivatives as the rate of change of a function, focusing on how dependent variable 'Y' changes with the independent variable 'X'. It explains differentiation, the process of finding the derivative, and clarifies the misconception of setting 'delta X' to zero. The script uses the example of a quadratic function to illustrate the calculation of the derivative and touches on the importance of considering both positive and negative 'delta X' values. It also introduces the absolute value function as a teaser for the next lesson.

Takeaways

• 📈 The derivative of a function measures the rate of change of a dependent variable 'Y' with respect to an independent variable 'X'.
• 🔄 Differentiation is the process of finding the derivative, which represents the instantaneous rate of change at a specific point.
• 📚 The average rate of change between two points is the slope of the secant line connecting those points on a graph.
• 🔍 As the interval between 'X' values decreases, the secant lines approach the tangent line, representing the instantaneous rate of change.
• 🚫 The limit as 'delta X' approaches zero does not mean setting 'delta X' to zero in the ratio, as this would result in an undefined 'zero over zero' expression.
• 🔑 The derivative is denoted by a dash (') placed over the function notation, indicating the instantaneous rate of change at a given 'X'.
• 📚 The derivative of the square of 'X' (Y = X^2) at a particular 'X' is found by considering the average rate of change and taking the limit as 'delta X' approaches zero.
• 🤔 The concept of 'delta X' tending to zero is crucial; it implies considering values of 'delta X' that are close to zero but never actually zero.
• 🔄 It's important to consider both cases when 'delta X' is greater than zero and when it is less than zero to find the derivative at a particular 'X'.
• 🤷‍♂️ The absolute value function presents a challenge for finding the derivative at 'X' equal to zero, as it requires separate consideration of the left and right limits.
• 📝 The script encourages viewers to think about the derivative of the absolute value function and to look forward to the next lesson for the solution.

Q & A

What does the derivative of a function measure?

The derivative of a function measures the rate of change of a dependent variable 'Y' with respect to an independent variable 'X' at a particular value of 'X'.

What is the process of finding the derivative of a function called?

The process of finding the derivative of a function is called differentiation.

What is the significance of the ratio in the context of differentiation?

The ratio represents the average rate of change between two values of 'X' and is the slope of the secant line between those two points on the function.

How does the average rate of change approach the instantaneous rate of change as delta X approaches zero?

As delta X approaches zero, the secant lines between points on the function get closer and closer to the tangent line at 'X not', which represents the instantaneous rate of change.

Why is it incorrect to substitute delta X with zero in the ratio of average rate of change?

Substituting delta X with zero would result in a division by zero, which is undefined and does not make sense in mathematics.

What does the notation with a dash above the function notation represent?

The notation with a dash above the function, such as f'(x), represents the derivative of the function at a particular value of 'X'.

In the example given, what is the function 'Y' in terms of 'X'?

In the example, 'Y' is the square of 'X', which can be written as Y = X^2.

How is the derivative of the square function found?

The derivative of the square function is found by considering the average rate of change as delta X tends to zero and simplifying the expression to get 2 * X at a particular value of 'X'.

Why is it necessary to consider both cases when delta X is greater than zero and less than zero?

It is necessary to consider both cases to ensure that the average rates approach the same limit as delta X tends to zero, which confirms the derivative of the function at that particular value of 'X'.

What is the absolute value or modulus function, and how does it relate to the derivative?

The absolute value or modulus function is defined as Y = |X|, where Y is the non-negative value of X. It is used to illustrate that finding the derivative at certain points, like X = 0, may require special consideration.

What is the conclusion about the derivative of the square function at a particular value of 'X'?

The derivative of the square function at a particular value of 'X' is 'two times X', which is found by considering the average rate of change for both positive and negative increments of 'X'.

Outlines

00:00

📚 Understanding the Derivative and Differentiation

This paragraph explains the concept of a derivative as a measure of the rate of change of a function. It introduces the dependent variable 'Y' and the independent variable 'X', and how 'Y' is a function of 'X'. The derivative at a specific 'X' value indicates the instantaneous rate of change of 'Y' with respect to 'X'. The process of finding the derivative is called differentiation. The paragraph also discusses the concept of average rate of change and how it approaches the instantaneous rate as the interval 'delta X' approaches zero. It clarifies that setting 'delta X' to zero in the ratio is not the same as taking the limit as 'delta X' approaches zero, which is a common misunderstanding. The example of the square of 'X' is used to illustrate the process of finding the derivative and the importance of considering the limit properly.

05:02

🔍 Deep Dive into the Derivative's Limit Concept

This paragraph delves deeper into the concept of limits in the context of derivatives. It emphasizes that 'delta X' should never actually be zero, but should approach zero to find the derivative. The paragraph uses the example of the derivative of the square of 'X' to demonstrate that the average rate of change converges to '2 times X one' as 'delta X' gets smaller. It also points out that considering 'delta X' as both positive and negative is crucial for finding the derivative, as it ensures the average rate of change approaches the same limit from both directions. The absolute value function is introduced as a challenge for the audience to consider its derivative at 'X' equal to zero, setting up for the next lesson. The paragraph concludes with an invitation for viewers to subscribe for more educational content.

Keywords

💡Derivative

The derivative of a function is a fundamental concept in calculus, representing the rate at which the output of a function changes with respect to changes in its input. In the video, the derivative is described as the instantaneous rate of change of the dependent variable 'Y' with respect to the independent variable 'X', which is crucial for understanding how quickly 'Y' changes as 'X' varies. The script uses the process of differentiation to find the derivative, emphasizing the importance of the limit as 'delta X' approaches zero.

💡Rate of Change

The rate of change is a measure of how quickly a quantity changes in relation to another. In the context of the video, it specifically refers to the speed at which 'Y' changes as 'X' changes. The script explains that the derivative at a particular value of 'X' quantifies this rate of change, indicating whether 'Y' is increasing or decreasing rapidly or slowly at that point.

💡Dependent Variable

In the script, the dependent variable, denoted by 'Y', is the quantity that changes in response to changes in another variable, 'X'. The video explains that 'Y' is a function of 'X', meaning its value is determined by the value of 'X'. The concept is central to understanding how functions operate and how their rates of change can be measured.

💡Independent Variable

The independent variable, represented by 'X' in the script, is the variable that can change freely and is not determined by any other variable within the function. It is the control for the dependent variable 'Y'. The video uses 'X' to illustrate how changes in the independent variable can be analyzed to determine the behavior of the dependent variable.

💡Function

A function, denoted by 'F' in the script, is a mathematical relationship that expresses 'Y' as a dependent variable in terms of 'X' as the independent variable. The video describes how the function dictates the changes in 'Y' based on the value of 'X', and the derivative of this function reveals the rate at which 'Y' changes with respect to 'X'.

💡Differentiation

Differentiation is the process of finding the derivative of a function. The script outlines this process, starting with the calculation of the average rate of change and then taking the limit as 'delta X' approaches zero to find the instantaneous rate of change. Differentiation is key to understanding the behavior of functions and their rates of change.

💡Average Rate of Change

The average rate of change is the ratio of the change in the dependent variable 'Y' to the change in the independent variable 'X' over a specific interval. The script explains how this ratio is used to find the slope of the secant line between two points on the function and how it approaches the slope of the tangent line as 'delta X' gets smaller, leading to the derivative.

💡Secant Line

A secant line in the script refers to a straight line that intersects a curve at two points. It is used to approximate the rate of change between these two points. As 'delta X' approaches zero, the secant lines become closer to the tangent line, which represents the instantaneous rate of change at a specific point on the curve.

💡Tangent Line

The tangent line is a straight line that touches a curve at a single point without crossing it. In the script, the tangent line is the limit of the secant line as 'delta X' approaches zero, representing the instantaneous rate of change of the function at that point. It is a critical concept in understanding the derivative.

💡Limit

The limit is a fundamental concept in calculus that describes the value that a function or sequence approaches as the input approaches a certain value. In the context of the video, the limit is used to define the derivative by considering what happens as 'delta X' approaches zero, thus finding the instantaneous rate of change.

💡Absolute Value

The absolute value, or modulus, of a number is the non-negative value of that number without regard to its sign. In the script, the absolute value function is introduced as an example where the derivative at 'X' equal to zero requires special consideration, highlighting the importance of understanding the behavior of functions at specific points.

Highlights

Derivative of a function measures the rate of change of a dependent variable 'Y' with respect to an independent variable 'X'.

The derivative at a specific 'X' value indicates how fast or slow 'Y' changes at that point.

Differentiation is the process of finding the derivative of a function.

The average rate of change is the ratio of the change in 'Y' to the change in 'X' between two points.

As 'delta X' approaches zero, the average rate of change approaches the instantaneous rate of change, which is the derivative.

The derivative is denoted by a dash placed on the function notation.

Substituting 'delta X' with zero in the derivative formula is a convenient way to find the limit as 'delta X' approaches zero.

The limit process does not involve setting 'delta X' equal to zero in the ratio, as this would result in an undefined expression.

The derivative of 'Y = X^2' at 'X = X1' is found by considering the average rate of change for 'delta X' approaching zero.

The derivative of the square function is '2 * X1', which is the instantaneous rate of change at 'X1'.

Substituting 'delta X' with zero in intermediate steps of differentiation is incorrect and can lead to confusion.

The correct approach is to consider 'delta X' tending towards zero to find the derivative, not setting it to zero explicitly.

The derivative must be considered for both 'delta X' greater than zero and less than zero to ensure consistency.

The absolute value or modulus function has a unique behavior where the derivative changes based on the sign of 'X'.

Finding the derivative of the absolute value function at 'X = 0' requires special consideration.

The derivative of a function at a particular 'X' is the value to which the average rate of change converges as 'delta X' approaches zero from both sides.

The video concludes with an invitation to find the derivative of the absolute value function at 'X = 0' and a prompt to subscribe for the next lesson.

Transcripts

Derivative of a function measures its ‘RATE of change’.

Imagine a quantity denoted by variable ‘Y’

which is continuously changing.

But how it changes is CONTROLLED by another

quantity, denoted by variable ‘X’.

We can say that the variable ‘Y’ called

the dependent variable is a FUNCTION of the

variable ‘X’ called the independent variable.

The function ‘F’ tells us how the value

of ‘Y’ changes with the value of ‘X’.

The derivative of a function at a particular

value of ‘X’, tells us the RATE of change

of ‘Y’ with respect to ‘X’, at that particular value of ’X’.

That is how fast or slow the value of ‘Y’

changes with respect to ‘X’.

In our previous video, we saw how to find

the derivative of a function.

The whole process is summarised like this.

It is called Differentiation.

This ratio here is called the average rate

of change between two values of ‘X’ which

are ‘X not’ and ‘X not plus delta X’.

If ‘delta X’ is greater than Zero, then

‘X not plus delta X’ will be somewhere

here on the ‘X axis’.

Then this ratio is the slope of this secant

line between these two points.

Now as we find the average rate in the interval

closer and closer to ‘X not’, we see that

these secant lines approach the tangent line at ‘X not’.

This would be true even if we take ‘Delta

X’ to be less than zero.

Then ‘X not plus delta X’ will be somewhere

here on the ‘X axis’.

And as we find the average rate in the interval

closer and closer to ‘X not’, we see that

these secant lines approach the tangent line at ‘X not’.

So we say that in the limit delta X tends

to Zero the average rate of change APPROACHES

the instantaneous rate of change at X not.

That is nothing but the derivative of the

function at X not.

We denote it by putting a dash like this on

the notation for the function.

Note that the limit delta X tends to zero

does not mean we put ‘Delta X’ equal to

zero in this ratio.

This would result in the ratio being equal

to ‘zero divided by zero’ which does not

make any sense.

In this video, we will understand what this

really means.

Along with it we will also explore different

theoretical aspects of the derivative of a

function.

Consider this simple example.

‘Y’ is equal to square of ‘X’.

Let’s say we want to find the derivative

of this function at a particular value of

‘X’, say ‘X one’.

Can you find this out?

Actually, in one of our previous videos, we

found the answer to this.

We had found the instantaneous speed of an

object for such a relationship between the

distance travelled and the time elapsed.

And to find the derivative we would first

find the average rate of change.

That is, how the value of function changes

as the value of ‘X’ changes by ‘delta

X’.

‘F of X one plus delta X’ and ‘F of

X one’ will be equal to this.

After expanding this term and simplifying,

we will get this.

And after dividing by ‘delta X’ we will

get this final expression.

Now we know that to get the derivative, we

will need to consider the interval ‘delta

X’ tending to zero.

So we put delta X equal to Zero here and get this.

It is the derivative of the function at X one.

But now notice that all these expressions

are equivalent to each other.

So what if instead of putting ‘delta X equal

to Zero’ here, we put ‘delta X equal to

Zero here or here?

We see that we will get the numerator and

the denominator both equal to zero.

But for this final expression, we got the

answer as two times X one.

So what is going on here?

Why can we substitute ‘delta X equal to

zero’ here and not here?

Actually, substituting ‘Delta X’ equal

to zero at any step here is not correct.

Notice that in these steps we are dividing

by delta X.

And we know we can’t divide by ‘Zero’.

What ‘delta X’ tends to zero means is

that we are considering smaller and smaller

values of ‘Delta X’.

The values of ‘Delta X’ are close to Zero,

but NEVER zero.

But among all these steps here, delta X does

not explicitly occur in the denominator here.

So It is easy to see from this step that as

delta X gets smaller and smaller, the average

rate gets closer and closer to ‘2 times X one’.

So substituting ‘Delta X’ equal zero here

is just CONVENIENT to reach this conclusion.

But always be aware of what it really means

to put ‘Delta X’ equal to zero here.

Now notice one more thing.

Although we had not explicitly stated it,

here we took delta X to be greater than zero.

That is, we are finding the average rate between

X one and values of 'X' greater than it.

But it is necessary to consider the case when

delta X is less than zero.

That is, finding the average rate between

X one and values of 'X' lesser than it.

As we find the average rate when delta X tends

to zero, this average rate should approach

‘two times X one’.

So let’s find the average rate in this case

when delta X is less than zero.

We can see the calculation for finding the

average rate of change will be the same as

this.

But in the first case, this average rate will

always be greater than 'two times X one'.

And in the second case this average rate will

always be less than 'two times X one'.

Now when delta X tends to zero, both these

average rates will approach the same limit

‘two times X one’.

Now we can conclude that the derivative of

this function at X one is equal 'two times

X one'.

So we see that to find the derivative of a

function at a particular value of ‘X’,

we have to find the average rate for two cases:

When ‘delta X’ is greater than zero and

when ‘delta X’ is less than zero.

Now when delta x tends to zero, if both these

average rates approach the same number, then

that number is the derivative of the function

at that particular value of ‘x’.

But consider this function now.

It is called the absolute value or modulus function.

These vertical bars say that the value of

‘Y’ is equal to only the non negative

value of X.

That is if ‘X’ is greater than or equal

to zero, then ‘Y’ is equal to ‘X’.

And if ‘X’ is less than zero then ‘Y’

is equal to ‘negative of X’.

Can you find the derivative of this function

at ‘X’ equal to zero?

Share your thoughts in the comments section below.

We will find its derivative in the next lesson.

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