Understanding Partial Derivatives: The Intuitive Approach
This video breaks down the concept of partial derivatives by connecting them to the familiar idea of ordinary derivatives. The key insight is that a partial derivative is just an ordinary derivative where you temporarily ignore all other variables.
The Core Concept: Nudging the Input
The video uses a powerful mental model to explain derivatives:
-
Ordinary Derivative (function of one variable):
- You have a function f(x).
- You take a tiny nudge in the x-direction, called $dx$.
- This causes a tiny change in the output, called $df$.
- The derivative $\frac{df}{dx}$ is the ratio of the output change ($df$) to the input nudge ($dx$). Graphically, this is the slope of the function at that point.
-
Partial Derivative (function of multiple variables):
- For a function like $f(x,y)$, your input space is a 2D plane, not a single line.
- $\frac{\partial f}{\partial x}$: You take a tiny nudge in the x-direction only ($dx$) and see how it changes the output ($df$). The y-coordinate is held constant.
- $\frac{\partial f}{\partial y}$: You take a tiny nudge in the y-direction only ($dy$) and see how it changes the output ($df$). The x-coordinate is held constant.
Why the name "Partial"? Each derivative only tells part of the story. It describes how the function changes in one specific direction, ignoring all others.
The Computation: Treating Variables as Constants
The video emphasizes the practical rule: To compute a partial derivative, treat all other variables as constants and take an ordinary derivative.
Example: $f(x, y) = x^2 y + \sin(y)$
1. Evaluate at a Specific Point: $\frac{\partial f}{\partial x}$ at $(1, 2)$
- Since you're moving in the x-direction, $y$ is constant ($y = 2$).
- Plug in the constant first: $f(x) = x^2 \cdot 2 + \sin(2)$.
- Now take the ordinary derivative with respect to $x$: $\frac{d}{dx}(2x^2) = 4x$.
- Evaluate at $x = 1$: $4(1) = 4$.
2. Evaluate at a Specific Point: $\frac{\partial f}{\partial y}$ at $(1, 2)$
- Since you're moving in the y-direction, $x$ is constant ($x = 1$).
- Plug in the constant first: $f(y) = 1^2 \cdot y + \sin(y) = y + \sin(y)$.
- Take the ordinary derivative with respect to $y$: $1 + \cos(y)$.
- Evaluate at $y = 2$: $1 + \cos(2)$.
3. Find the General Formula
-
$\frac{\partial f}{\partial x}$ as a function of $x$ and $y$:
- Treat $y$ as a constant.
- $\frac{\partial}{\partial x}(x^2 y) = 2xy$ (derivative of $x^2$ times the constant $y$).
- $\frac{\partial}{\partial x}(\sin(y)) = 0$ (derivative of a constant is 0).
- Result: $\frac{\partial f}{\partial x} = 2xy$
-
$\frac{\partial f}{\partial y}$ as a function of $x$ and $y$:
- Treat $x$ as a constant.
- $\frac{\partial}{\partial y}(x^2 y) = x^2$ (derivative of $y$ times the constant $x^2$).
- $\frac{\partial}{\partial y}(\sin(y)) = \cos(y)$.
- Result: $\frac{\partial f}{\partial y} = x^2 + \cos(y)$
Key Takeaways
- Compute: To find $\frac{\partial f}{\partial x}$, treat $y$ as a constant and differentiate normally. To find $\frac{\partial f}{\partial y}$, treat $x$ as a constant and differentiate normally.
- Interpret: A partial derivative measures the instantaneous rate of change of a function as you move in one specific direction (the x-axis or y-axis), while keeping all other inputs fixed.
- Think Beyond Graphs: While you can visualize partial derivatives as slopes on a 3D surface, the core idea, nudging an input and measuring the output change, applies to functions with any number of dimensions, even when graphs aren't possible. This connects nicely to the broader Understanding First-Order Ordinary Differential Equations: Geometric and Numerical Approaches in mathematics, where the rate of change remains a central concept. For a deeper dive into the underlying theory for functions of several variables, the Understanding Cauchy-Riemann Relations and Holomorphic Functions page offers an excellent extension into complex analysis. Finally, a Comprehensive Formula Revision: Linear Algebra & Calculus Explained can help solidify these and other essential calculus concepts.
so let's say I have some multivariable function like f of XY so it'll have a two variable input is equal to I don't
know x^2 * y plus s of Y so it'll output just a single number it's a scalar valued function question is how do we
take the derivative of an expression like this and there's a certain method called a partial derivative which is
very similar to ordinary derivatives and I kind of want to show how they're secretly the same thing so to do that uh
let me just remind ourselves of how how we interpret the notation for ordinary derivatives so if you have something
like FX is equal x^2 and let's say you want to take its derivative and I'll use the libet
notation here DF DX and let's evaluate it at two let's say I really like this notation because
it's suggestive of what's going on if we sketch out a graph so you know this axis represents
our output this over here represents our input and X squ has a certain parabolic shape to it something like that um then
we go to the input x = 1 2 this little DX here I like to interpret as just a little nudge in the
X Direction and it's kind of the size of that nudge and then DF
DF is the resulting change in the output after you make that initial little nudge so it's this resulting change and when
you're thinking in terms of graphs this is slope you kind of have this rise over run for your ratio between the tiny
change of the output that's caused by a tiny change in the input and of course this is dependent on where you start
over here we have xal 2 um but you could also think about this without graphs if you really wanted to you might just
think about you know your input space is just a number line and your output space Also is just a number line the
output of f over here and really you're just thinking of somehow mapping numbers from here onto the second line and in
that case your initial nudge your initial little DX would be some nudge on that number line and you're wondering
how that influences the the function itself so maybe that causes a nudge that's you know four times as big and
that would mean your derivative is uh four at that point so the reason that I'm talking about this is because over
in the multivariable world we can pretty much do the same thing you know you could write DF
DX and interpret that as saying hey how does a tiny change in the input in the X Direction influence the output but this
time the way that you might visualize it you'd be thinking of your input space here I'll I'll draw it down here as the
XY plane so this time this is not going to be graphing the function this is every point on the plane is an input and
let's say you were evaluating this at a point like one two okay um in that case so You' go over
to the input that's one and then two and then you'd say Okay so this tiny nudge in the input this tiny change DX
how does that influence the output and in this case the output I mean it's still just a number so maybe we go off
to the side here and we draw just like a number line as our output and somehow we're thinking about
the function as mapping points on the plane to the number line so You' say okay that's your that's your DX how much
does it change the output and you know maybe this time it changes it negatively it depends on your function and that
would be your DF and you can also do this with the Y variable right there's no reason to you can't say DF
Dy and evaluated that same point 1 2 and interpret totally the same way except this time your
Dy would be a change in the y direction so maybe I should really emphasize here that that DX that that DX
is a change in the X Direction here and that Dy is a change in the y direction and maybe when you change your F
according to Y it does something different right maybe you know uh the output increases and it increases
by a lot it's more sensitive to Y again it depends on the function um and I'll show you how you can compute something
like this in in just a moment here but first there's kind of an annoying thing associated with partial derivatives
where we don't write them with D's in DX DF uh people came up with this new notation mostly just to emphasize to the
reader of your equation that it's a multivariable function involved and what you do is you say uh you write a d but
it's got kind of a curl at the top it's this new symbol and people will often read it as partial so you might read
like partial F partial y if you're wondering by the way why we call these partial derivatives it's sort of like
this doesn't tell the full story of how F changes because it only cares about the X Direction neither does this this
only cares about the Y Direction so each one is only um a small part of the story so let's actually evaluate something
like this I'm going to go ahead and clear the board over here I think the one-dimensional
analogy and something we probably have already so little
remnants so if you're actually evaluating something like this here I'll write it again up here partial
derivative of f with respect to X and we're doing it at 1 two it only cares about movement in the X Direction so
it's treating y as a constant it doesn't even care about the fact that y changes as far as it's concerned Y is always
equal to two so we can just plug that in ahead of time so I'm going to say partial partial X this is another way
you might write it put the expression in here and I'll say x^ s but instead of writing y I'm just going to plug in that
constant ahead of time because when you're only moving in the X Direction this is kind of how how
the multivariable function sees the world and I'll just keep a little note that we're evaluating this whole thing
at xals 1 and here this is actually just an ordinary derivative right this this is an expression that's an X you're
asking how it changes as you shift around X and you know how to do do this this is just taking the taking the
derivative um the derivative of uh x^2 * 2 is going to be 4X because x^2 goes to 2X and then the derivative of a constant
s of two is just a constant is zero and of course we're evaluating this at xal 1 so your overall answer is going
to be four and just for practice let's also do that with the derivative with respect to Y so we look over here I'm
going to write the same thing you're taking the partial derivative of f with respect to
y we're evaluating it at the same point 1 2 this time it doesn't care about movement in the X
Direction so as far as it's concerned that X just stays constant at 1 so we'd write 1^ 2 * y plus s of
y s of Y and you're saying oh I'm keeping track of this at yal 2 so that's kind of
you're evaluating at y equal 2 when you take the derivative this is just 1 * y so the derivative is 1 this over here
the derivative is cosine cosine of Y again we're evaluating this whole
thing at yal 2 so your overall answer it would be you know 1 plus cosine of 2 I'm not sure I'm not sure what the value of
cosine of two is off the top of my head but that would be your answer and this is this is a partial
derivative at a point but a lot of times you're not asked to just computed at a point what you want is a general formula
that tells you hey plug in any point XY and it should spit out the answer so let me just kind of go over how you would do
that it's actually very similar but this time instead of plugging in the constant ahead of time we just have to pretend
that it's a constant so let me um let me make a little bit of space for ourselves here
really we don't need any of this anymore I'm going to leave the partial partial F partial partial y
we want this as a more General function of X and Y well we kind of do the same thing
we're going to say that this is you know derivative with respect to X and I'm using partials just to kind of emphasize
that it's a partial derivative but now we'd write x^2 and then kind of emphasize that it's a constant value of
y plus the sign and again I'll say Y and here I'm writing the variable y but we have
to pretend like it's a constant you're pretending that you plug in two or something like that and you still just
take the derivative so in this case the derivative of x^2 * a constant is just 2x * that constant 2x times that
constant and over here the derivative of a constant is always zero so that's just always going to be zero so this is your
partial derivative as a more general formula if you plugged in one two to this you'd get what we had before and
simil similarly if you're doing this with f of uh partial F partial y we write down all of the same things
now you're taking it with respect to Y and I'm just going to copy this formula here
actually but this time we're considering all of the X's to be constants so in this case when you take the derivative
with respect to Y of some kind of constant you know constant squ is a constant time y it's just going to equal
that constant so this is going to be x^2 and over here you're taking the derivative of s of Y there's no x's in
there so that remains the S of Y clear and now this is a more general
formula if you plugged in one two you would get one so oh sorry that's cosine of
Y cosine of Y because we're taking a derivative um so if you plugged in one two you know
you would get 1 plus the cosine of one which is what we had before so this this is really what you'll see for how to
compute a partial derivative you pretend that one of the variables is constant and you take an ordinary derivative and
in the back of your mind you're thinking this is because you're just moving in One Direction uh for the input and
you're seeing how that influences things and then you know you might move in one direction for another input and see how
that influences things in the next video I'll show you what this means in terms of graphs and slopes but it's important
to understand that graphs and slopes are not the only way to understand derivatives because as soon as you start
thinking about Vector valued functions or functions with inputs of higher Dimensions than just two you can no
longer think in terms of graphs but this idea of nudging the input in some direction seeing how that influences the
output and then taking the ratio you know the ratio of that output nudge to the input nudge that's a more General
way of viewing things and that's going to be very helpful moving forward in multivariable C
An ordinary derivative measures how a function changes when you nudge a single input variable along a line. A partial derivative does the same, but for a function with multiple variables—you nudge just one input direction (like x or y) and hold all other variables constant, so it tells only 'part' of the function's sensitivity to that specific direction.
Simplify the process by treating every variable except the one you're differentiating with respect to as a constant. Then apply the standard rules of ordinary differentiation. For example, to find ∂f/∂x for f(x, y) = x²y + sin(y), treat y as constant: the derivative of x²y becomes 2xy, and sin(y) becomes 0.
It's called 'partial' because each derivative reveals only one piece of the overall change—how the function responds to movement along a single axis (e.g., x) while freezing the other variables. To understand the full sensitivity, you need all partial derivatives together, forming the gradient.
Yes. First plug in the constant value for the other variable(s), then differentiate normally with respect to the variable of interest. For f(x, y) = x²y + sin(y) at (1,2): to find ∂f/∂x, substitute y=2 to get f(x)=2x²+sin(2), then differentiate to get 4x, and evaluate at x=1 to obtain 4.
Geometrically, a partial derivative like ∂f/∂x at a point gives the slope of the tangent line to the 3D surface f(x,y) when you slice the graph parallel to the x-axis (holding y fixed). It tells you how steeply the function rises or falls as you move east/west, ignoring any north/south movement.
Absolutely. The core idea—nudge one input, keep all others fixed, measure the output change—works for functions of any number of variables, even when a graph is impossible to visualize. This makes partial derivatives essential for fields like physics, economics, and machine learning, where models often have many inputs.
Partial derivatives are fundamental to understanding rates of change in multivariable systems, which connects directly to first-order ordinary differential equations (where the rate of change is central). They also underpin advanced concepts like holomorphic functions (via Cauchy-Riemann relations) and appear in comprehensive formula revisions that link linear algebra and calculus.
Keep this summary
Save it to LunaNotes and it becomes a real note in your library — editable, searchable, and ready to turn into flashcards or a diagram. Free to start.
Save to LunaNotesOr summarise for another video.
This summary and transcript were automatically generated using AI with the Free YouTube Transcript Summary Tool by LunaNotes.
Related summaries
Understanding First-Order Ordinary Differential Equations: Geometric and Numerical Approaches
This lecture explores first-order ordinary differential equations (ODEs) focusing on geometric interpretations and numerical methods. Learn about direction fields, integral curves, isoclines, and the Existence and Uniqueness Theorem, with practical examples illustrating solution behaviors and limitations.
Understanding Cauchy’s Theorem and Complex Integrals Explained
This video breaks down the fundamentals of complex integrals and provides a clear, step-by-step proof of Cauchy's theorem—a cornerstone in complex calculus. Learn how holomorphic functions behave on closed curves and why their contour integrals vanish under specific conditions.
Understanding Rectangular and Polar Coordinates for Advanced Function Analysis
Explore the complexities of rectangular and polar coordinates, their differences, and functional behavior in advanced mathematics.
Understanding Cauchy-Riemann Relations and Holomorphic Functions
This video explains the essential Cauchy-Riemann relations that determine whether a complex function is holomorphic (complex differentiable). It explores two examples to illustrate why one function meets these conditions while another does not, and highlights powerful theorems regarding infinite differentiability and Taylor expansions of holomorphic functions.
Introduction to Functions of Complex Variables and Holomorphicity
This lecture introduces the fundamental concepts of functions of complex variables, focusing on their definition, differentiability, and the special class of holomorphic functions. Through examples such as the squaring function and complex polynomials, it highlights the stringent criteria for complex differentiability and distinguishes holomorphic functions from non-differentiable ones.
Most viewed summaries
A Comprehensive Guide to Using Stable Diffusion Forge UI
Explore the Stable Diffusion Forge UI, customizable settings, models, and more to enhance your image generation experience.
Kolonyalismo at Imperyalismo: Ang Kasaysayan ng Pagsakop sa Pilipinas
Tuklasin ang kasaysayan ng kolonyalismo at imperyalismo sa Pilipinas sa pamamagitan ni Ferdinand Magellan.
Mastering Inpainting with Stable Diffusion: Fix Mistakes and Enhance Your Images
Learn to fix mistakes and enhance images with Stable Diffusion's inpainting features effectively.
Pamamaraan at Patakarang Kolonyal ng mga Espanyol sa Pilipinas
Tuklasin ang mga pamamaraan at patakaran ng mga Espanyol sa Pilipinas, at ang epekto nito sa mga Pilipino.
How to Install and Configure Forge: A New Stable Diffusion Web UI
Learn to install and configure the new Forge web UI for Stable Diffusion, with tips on models and settings.
Found this summary useful?
Take it with you. One click puts it in your own LunaNotes library.
Save to LunaNotes