Derivative definition
Summary
TLDRThe 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
🌱 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.
📚 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.
Mindmap
Keywords
💡Derivative
💡Single Variable Function
💡Notation
💡Instantaneous Rate of Change
💡Limit
💡Delta X (Δx)
💡F Prime (f')
💡Leibniz Notation
💡Operator Notation
💡Variable
💡Derivative Function
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
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