Applying First Principles to x² (1 of 2: Finding the Derivative)
Summary
TLDRThe video script explains the concept of a derivative in calculus, focusing on how the gradient of a curve changes and cannot be represented by a single constant. It introduces the gradient as a function, leading to the concept of the derivative, which is a function itself. The script discusses the process of finding the derivative from first principles using the limit of the difference quotient. The explanation emphasizes understanding the concept rather than rote memorization and applies it to a specific example involving the function f(x) = x².
Takeaways
- 📚 The concept of 'rise over run' is revisited in the context of a curve where the gradient is not constant but changes, leading to the idea of a gradient function.
- 🔍 The term 'derivative' is introduced as the rate of change of a function, symbolized as f'(x) or \( \frac{dy}{dx} \), which represents the gradient function.
- 📈 The process of finding the derivative involves taking the limit of the rise over run as the run approaches zero, which is the definition of the derivative.
- 👉 The notation f'(x) is used when the function is named 'f', while \( \frac{dy}{dx} \) is used in more general contexts without a specific function name.
- 🚫 Memorizing the process without understanding is discouraged; the importance of grasping the underlying concepts is emphasized.
- 🔑 The derivative is calculated by finding the limit as \( h \) approaches zero in the expression \( \frac{f(x+h) - f(x)}{h} \).
- 📉 The example of the parabola f(x) = x^2 is used to demonstrate the process of finding the derivative from first principles.
- 🔄 The process involves algebraic manipulation to simplify the expression and isolate the variable \( h \) to cancel it out before taking the limit.
- 🚫 The limit cannot be taken at \( h = 0 \) directly due to division by zero, but the behavior as \( h \) approaches zero is considered.
- 📌 The derivative of f(x) = x^2 is found to be 2x, which represents the slope of the tangent line to the curve at any point.
- 🔍 The concept of a 'hole' in the derivative function at x = -1 is discussed, indicating a point where the derivative does not exist.
Q & A
What is the concept of 'rise over run' in the context of a curve with changing gradient?
-In the context of a curve with a changing gradient, 'rise over run' is not a constant but a function itself. It represents the change in y (rise) over the change in x (run), and is expressed as dy/dx, which is the gradient function of the curve.
Why can't we use the term 'gradient' in the traditional sense for a curve with a variable slope?
-We can't use the term 'gradient' in the traditional sense because for a curve with a variable slope, the gradient is not a constant value; it changes at every point on the curve, hence it is better described as a gradient function.
What is the term used to describe the gradient of a function?
-The term used to describe the gradient of a function is 'derivative'. It signifies that the gradient is derived from the original function and is itself a function.
How is the derivative of a function represented mathematically?
-The derivative of a function is represented mathematically as f'(x), which is another way of indicating the notation for the gradient function of f(x).
What is the significance of the limit as h approaches zero in the context of derivatives?
-The limit as h approaches zero is used to find the derivative of a function at a specific point. It helps in transitioning from the concept of the gradient between two points (secant) to the gradient at a single point (tangent).
Why is it important to understand the origin of mathematical concepts like derivatives?
-Understanding the origin of mathematical concepts like derivatives is crucial for true comprehension. It prevents mere memorization without grasping the underlying principles, which is essential for applying these concepts effectively.
What is the difference between the gradient of a tangent and the gradient of a secant?
-The gradient of a tangent is the instantaneous rate of change at a specific point on a curve, while the gradient of a secant is the average rate of change between two points on the curve. The limit as h approaches zero is used to find the tangent's gradient, which is the derivative.
What is the process of finding the derivative of a function from first principles?
-The process involves taking the limit of the difference quotient (f(x+h) - f(x)) / h as h approaches zero. This manipulation helps in isolating h and finding the derivative at a particular point on the function.
Can you provide an example of finding the derivative of a simple function, like f(x) = x^2?
-Yes, for f(x) = x^2, the derivative f'(x) is found by taking the limit as h approaches zero of (x+h)^2 - x^2 / h, which simplifies to 2x after canceling out terms and applying the limit.
Why is there a hole in the graph of the function (x^2 - x^2) / h as h approaches zero?
-There is a hole at x = -1 because when h approaches zero, the expression (x+h)^2 - x^2 simplifies to 2x + h, and when x = -1, the term 2x + h becomes zero, leading to division by zero, which is undefined.
How does the concept of limits help in understanding the behavior of a function at a point where direct calculation is not possible?
-The concept of limits allows us to understand the behavior of a function as it approaches a certain point, even when direct calculation is not possible due to division by zero or other undefined operations. It provides a meaningful result by showing the trend or value the function is approaching.
Outlines
📚 Introduction to the Derivative Concept
This paragraph introduces the concept of the derivative in calculus, emphasizing the shift from thinking of the gradient as a constant to recognizing it as a variable function. It explains the notation \( \frac{d y}{d x} \) as representing the change in y over the change in x, and introduces the derivative as the gradient function. The paragraph also discusses the first principles approach to finding the derivative, using the limit as \( h \) approaches zero to differentiate between the tangent and secant lines. The importance of understanding the conceptual basis of derivatives rather than just memorizing formulas is highlighted.
🔍 Calculating the Derivative of a Parabola
The second paragraph delves into the process of calculating the derivative of the function \( f(x) = x^2 \) from first principles. It begins by setting up the limit notation and substituting the function into the derivative formula. The paragraph then walks through the algebraic manipulation required to simplify the expression, including factoring out a common \( h \) and canceling terms to isolate the variable. The discussion touches on the significance of the limit as \( h \) approaches zero and the conceptual understanding of approaching a value without actually reaching it, which is key to grasping the derivative's meaning. The final result of the derivative for \( f(x) = x^2 \) is \( 2x \), which is derived by recognizing the pattern in the simplified expression and applying the limit concept.
Mindmap
Keywords
💡Rise over run
💡Gradient function
💡Derivative
💡Limit
💡First principles
💡Secant
💡Tangent
💡Parabolas
💡Factorization
💡Instantaneous rate of change
Highlights
Introduction of the concept of gradient as a function, not a constant, in the context of a curve with changing slope.
Explanation of the notation d/dx for the derivative, representing the change in y over the change in x.
Clarification on the term 'derivative' as it relates to the gradient function derived from the original function.
Emphasis on understanding the origin of mathematical concepts rather than just memorizing them.
Discussion on the difference between the gradient of the tangent and the secant, highlighting the importance of the limit as h approaches zero.
Illustration of the process to find the derivative of a function from first principles, using the limit definition.
Demonstration of the derivative calculation for a simple function f(x) = x^2, emphasizing the steps and rationale.
The significance of the common factor of h in the numerator for simplifying the derivative expression.
Explanation of why h cannot be zero in the derivative calculation and the concept of approaching a limit.
The final simplification of the derivative of f(x) = x^2 to 2x, showcasing the result of the limit process.
Introduction of the concept of a hole in the derivative function at x = -1 due to the division by zero.
Discussion on the practical implications of a hole in the derivative function and its mathematical significance.
The importance of recognizing the approach to a limit even when the exact value cannot be calculated.
Reinforcement of the conceptual understanding of the tangent versus the secant in the context of derivatives.
The final expression of the derivative as 2x, highlighting the outcome of the limit process and its meaning.
Reflection on the process of deriving the derivative and the importance of understanding each step.
Encouragement for students to internalize the concept of derivatives and their application to functions.
Transcripts
picking up where we left off from
yesterday
what we finished with was thinking about
rise over run
in the specific context where you have a
curve like this and its grain is
changing all the time right so gradient
like we usually with straight lines
corner junction and so on we say oh
gradient
we'll just call it m
right because it's just a number it's
just a constant it's not changing it's
not variable right
but here m is not going to cut it m is
not going to cut because in fact the
gradient is not a number it's not a
constant it is a function itself the
gradient is changing wherever you look
right
so therefore we replace this idea of the
gradient as a
constant with the gradient as a function
in fact we call it the gradient function
right and rather than say rise over run
which is just kind of a nice mnemonic
way of remembering what gradient is we
said rise is really the change d delta
the change in y
and run is really the change in x and
that's where we get this d y over dx
notation from okay
now one word i didn't introduce
yesterday is that we call this thing
not just the gradient right but now that
it's a function it's its own thing we
call it the derivative
and derivative you know it just means it
comes from something else right namely
it comes from the actual function right
we learnt this way of taking this rise
over run and putting on these different
points and comparing it with limits
to what was happening at a particular
point on there okay so this is the
introduction the
actual thing that we said was first
principles right we said f dash which is
just yet another way of indicating the
notation right is equal to
the limit
as h approaches zero of now what was our
numerator what was it
f of x plus h f of x plus h right and
then we take away
f of x now just remember why do we do
that it's because if x plus h is here
and f of x is here and we just want the
difference that's all it is that's the
rise and of course there's the run
do you remember when i said to you we're
introducing this topic the worst thing
that you can do is just get get to work
memorizing stuff and not know where it
comes from um
i remember sitting next to you when i
was in year 11 sitting next to people
who just like look it's just just
memorize it okay doesn't matter what any
of it means okay
that's the quickest way to get on the
path of not actually understanding
what's happening
this is rise
right
this is run
one last thing what difference
does that make if it weren't there that
limit as h approaches zero if it weren't
there it's not meaningless but what
would it be it'd be something else yeah
it'll be the gradient of
not that not the tangent the tangent is
what we want right but if you've got two
points that are actually a part it'll be
the secant wouldn't right it's not
meaningless that's how we started it's
just not what we want this is the
gradient of the tangent the derivative
without that and many people like i have
to write this over and over again
without that it's the gradient of the
second we're not after that
okay so now that's where we stopped
right now we want to apply it to a
particular situation okay so here's a
function right now the one we'll start
with the easiest one to start with is
just the parabola x squared okay
if you see f of x right then you can use
this f dash x notation okay if on the
other hand
all you see is this
there is no function called f right so
none of this really makes sense f dash
what is any of that in here you would
have to use this notation up here right
if you've got y's you'd use d y and d x
if you've got f's you'd use
f and f dash okay
now i'm just going to come back because
for this example we're just starting off
i do actually want to go with um f
because i'll make this a bit easier so
this is f of x equals x squared
okay
let's proceed through this right if f of
x is that
then f dash x
is going to equal 2. now
it begins with just the substitution if
we know what f is i should know what all
those pieces are in there okay so
we begin with the limit
because i'm interested in the tangent
and not the secant
if f of x is x squared
then f of
x plus h
is x plus h all squared you agree with
that
and there's f of x just as we defined it
we're dividing through by h okay now
what we're about to do is what we call
evaluating the derivative evaluating the
derivative from first principles this is
our starting point okay
what we're trying to do is manipulate
this in such a way such that i can
actually put h equals 0 in there and see
what happens i can't do that right now
because h is on the denominator you
divide by zero it explodes okay so i
want to reshape this into something
that gets rid of h being just by itself
on the denominator okay so let's give
this a go obviously you look at the
numerator that's the only thing you can
do anything with here the numerator the
denominator is just simplified so i
write my limit
because i want the tangent not the
secant you're going to get really sick
of me saying that but i want you to get
in your head it's really important to
keep on saying it
on the numerator perfect square when you
expand it you get
x squared plus
2hx plus h squared and then there's that
minus x squared attacking along the end
okay and then of course everything is
divided by h right just because you're
not doing any work with them don't
forget to write this or the denominator
they're still there and they're still
critically important
having expanded you can see i can do
something with this now can't i got an x
squared on the front minus x squared on
the back
and they're going to cancel each other
out okay
now the reason why this is good is
because now i have a common factor of h
on the numerator right common factor
ratio let me take it out let me
factorize it
limit as h approaches 0 because i want
the tangent not the secant
of what's on the numerator
h outside of
2x
plus h good there's still there's still
an h hanging around there that's all
divided through by h okay
fantastic now i can cancel
denominator's gone
yep isn't that like the equivalent of
dividing like
by like like
canceling out
yes this is exactly right i'm glad you
raised that point why can i get rid of
this okay
let me pull back to you if you're here
like this um
this okay now what does this thing look
like okay uh
sorry yes
yes thank you
okay
now i was on autopilot for a second um
what does this thing look like even
though it's got next squared on the top
because of what you get on the bottom
you're actually going to get a straight
line aren't you okay when you factorize
x approaches negative one
um x minus one
so what does this thing actually look
like it looks like this straight line
with one difference namely
there's a hole there there's just a hole
right so
uh x minus one what does that look like
down here
and x equals negative one somewhere over
here i guess
there's my little hole there okay so our
problem is i can't simply input x equals
negative one because then as you notice
i'm i'm multiplying by zero and dividing
by zero and it loses meaning okay
however what i'm just come back to when
we define this idea of our limit right
what i'm trying to think about is what
am i
getting
towards
right and i am actually getting towards
something real even though i can't be
there itself any more than i can
calculate the gradient of you know a
point to itself that's what this is
really doing rise over run as the two
points get close together right
i can't actually calculate that but i
can still see what it's approaching and
if it's approaching the same thing from
both angles then that's fine that's
great i can take that as a meaningful
result okay
so though a very good point to mention
because it's like yeah why can't i do
that and the answer is because h can't
be actually equal to zero so i can take
it out of the equation
expression i should say now that i've
done this
this is the last time i'm going to write
the limit as h approaches zero because i
want say it with me
the tangent
not the secret right
just like a hammer anywhere okay so now
that i'm there
i've gotten rid of h on the denominator
all right
by the way h can be on the denominator
just not by itself because when it's by
itself the denominator becomes zero as
you'll see shortly
now i actually can't say well let's just
see what happens when h is zero and the
answer is it's two x plus zero don't
miss the plus zero it's not trivial okay
just like
multiplying by one sometimes is not
trivial adding zero is not trivial
because i see it comes from here and
that of course is just 2x okay
what's mean
تصفح المزيد من مقاطع الفيديو ذات الصلة
Applying First Principles to x² (2 of 2: What do we discover?)
Derivative definition
Derivative as a concept | Derivatives introduction | AP Calculus AB | Khan Academy
Calculus- Lesson 8 | Derivative of a Function | Don't Memorise
Differentiation (Maxima and Minima)
Jika lim x->-3 (x^2+4x+3)/(x+3)=a-1,nilai a adalah...
5.0 / 5 (0 votes)