Calculus- Lesson 8 | Derivative of a Function | Don't Memorise
Summary
TLDRThis 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
- đ The derivative of a function measures the rate of change of a dependent variable 'Y' with respect to an independent variable 'X'.
- đ Differentiation is the process of finding the derivative, which represents the instantaneous rate of change at a specific point.
- đ The average rate of change between two points is the slope of the secant line connecting those points on a graph.
- đ As the interval between 'X' values decreases, the secant lines approach the tangent line, representing the instantaneous rate of change.
- đ« 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.
- đ The derivative is denoted by a dash (') placed over the function notation, indicating the instantaneous rate of change at a given 'X'.
- đ 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.
- đ€ 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.
- đ 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'.
- đ€·ââïž 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.
- đ 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
đ 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.
đ 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.
Mindmap
Keywords
đĄDerivative
đĄRate of Change
đĄDependent Variable
đĄIndependent Variable
đĄFunction
đĄDifferentiation
đĄAverage Rate of Change
đĄSecant Line
đĄTangent Line
đĄLimit
đĄAbsolute Value
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 âRATE of changeâ.
Imagine a quantity denoted by variable âYâ
which is continuously changing.
But how it changes is CONTROLLED by another
quantity, denoted by variable âXâ.
We can say that the variable âYâ called
the dependent variable is a FUNCTION of the
variable âXâ called the independent variable.
The function âFâ tells us how the value
of âYâ changes with the value of âXâ.
The derivative of a function at a particular
value of âXâ, tells us the RATE of change
of âYâ with respect to âXâ, at that particular value of âXâ.
That is how fast or slow the value of âYâ
changes with respect to âXâ.
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 âXâ which
are âX notâ and âX not plus delta Xâ.
If âdelta Xâ is greater than Zero, then
âX not plus delta Xâ will be somewhere
here on the âX axisâ.
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 âX notâ, we see that
these secant lines approach the tangent line at âX notâ.
This would be true even if we take âDelta
Xâ to be less than zero.
Then âX not plus delta Xâ will be somewhere
here on the âX axisâ.
And as we find the average rate in the interval
closer and closer to âX notâ, we see that
these secant lines approach the tangent line at âX notâ.
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 âDelta Xâ equal to
zero in this ratio.
This would result in the ratio being equal
to âzero divided by zeroâ 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.
âYâ is equal to square of âXâ.
Letâs say we want to find the derivative
of this function at a particular value of
âXâ, say âX oneâ.
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 âXâ changes by âdelta
Xâ.
âF of X one plus delta Xâ and âF of
X oneâ will be equal to this.
After expanding this term and simplifying,
we will get this.
And after dividing by âdelta Xâ we will
get this final expression.
Now we know that to get the derivative, we
will need to consider the interval âdelta
Xâ 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 âdelta X equal
to Zeroâ here, we put âdelta 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 âdelta X equal to
zeroâ here and not here?
Actually, substituting âDelta Xâ 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 canât divide by âZeroâ.
What âdelta Xâ tends to zero means is
that we are considering smaller and smaller
values of âDelta Xâ.
The values of âDelta Xâ 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 â2 times X oneâ.
So substituting âDelta Xâ equal zero here
is just CONVENIENT to reach this conclusion.
But always be aware of what it really means
to put âDelta Xâ 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
âtwo times X oneâ.
So letâs 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
âtwo times X oneâ.
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 âXâ,
we have to find the average rate for two cases:
When âdelta Xâ is greater than zero and
when âdelta Xâ 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 âxâ.
But consider this function now.
It is called the absolute value or modulus function.
These vertical bars say that the value of
âYâ is equal to only the non negative
value of X.
That is if âXâ is greater than or equal
to zero, then âYâ is equal to âXâ.
And if âXâ is less than zero then âYâ
is equal to ânegative of Xâ.
Can you find the derivative of this function
at âXâ equal to zero?
Share your thoughts in the comments section below.
We will find its derivative in the next lesson.
Donât forget to subscribe in order to get notified.
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