Main content

### Course: Multivariable calculus > Unit 2

Lesson 1: Partial derivatives# Graphical understanding of partial derivatives

One of the best ways to think about partial derivatives is by slicing the graph of a multivariable function. Created by Grant Sanderson.

## Want to join the conversation?

- If the derivative of a constant*variable = constant how come in the first evaluation the partial derivative respect to x =>x²*y=2xy and in the second evaluation the partial derivative respect to y=>x²*y=x². I know that the power rule but don't understand why the place of the constant matters.(4 votes)
- But the place of the constant doesn't matter. In the first evaluation of partial derivative respect to x => x^2y = 2xy because we are considering y as constant, therefore you may replace y as some trivial number a, and x as variable, therefore derivative of x^2y is equivalent to derivative of x^2.a which is 2a.x , substitute trivial a with y and we have 2xy. In the latter case we considering x as trivial a and y as constant so we get 1.x^2(13 votes)

- I understand the video， but for only 2 variables，how can you have 3-dimention image(3 votes)
- not sure exactly what you mean but i'll try and answer.

If you mean you don't understand why we have a 3d image for a problem with 2d input it's because the 3rd dimension represents your output.

If you mean you get how this works for 2 variables but like to see how it looks for 3, we can't really graph that since our brain can't perceive a fourth dimension. I think this is what he was getting at, the graphical intuition is good but gets you only thus far with multivariable calculus because of the high dimensionality. There is still ways to represent such functions graphically, plotting input and output separately for example is what comes to mind, but I am just starting to learn calculus, so someone more advanced needs to answer.(10 votes)

- The equation in the video using only x and y but the graph also has the z-axis. So can we assume that the z was missing as part of the equation?(2 votes)
- z=f(x, y)

for every pair of (x, y) as the input, f(x,y) is the output, and the point at (x,y) with height f(x,y) is graphed.

https://www.khanacademy.org/math/multivariable-calculus/thinking-about-multivariable-function/ways-to-represent-multivariable-functions/a/multidimensional-graphs(8 votes)

- what software are you using to make the plot? what about the plane?(5 votes)
- I'm having hard time to visualize the partial derivative of a sphere,

if we for example took the y-axis as our constant plane, then we would have two different (z) values for a given (x) value, each of which has different slope (derivative).

As a representation for this part it would look like a circle, so the two different (z) values would be as follows : the upper value and the lower value; and they will have slopes that are opposite to each other in signs.

Hence, our partial derivative function should return 2 different answers, one for the upper (z) value and one for the lower (z) value. But this simply doesn't happen, can someone explain to me. It would be great if there were a graphical representation of this situation.

Thanks in advance :)(2 votes)- I don't know how well I could answer your question

As you said ,the part would look like a circle

Let's assume the equation of the sphere as z^2 +x^2+y^2=25

and let our constant y plane be y=2

So the circle's equation will be

z^2=25-x^2-(2)^2

z^2=21-x^2

Now the equation of the upper half of the circle will be

z=+sqrt(21-x^2)

and lower part of the circle will be

z=-sqrt(21-x^2)

So accordingly we use these two equations

So it doesn't return 2 different answers

Hope it helped a bit(3 votes)

- I'm confused about the pronunciations of multi, because if it's multivariable, people pronunce multi as mult-ee. But if it's multidemensional@5:48, he pronunced it as mult-eye.(1 vote)
- Pronunciations vary from person to person and region to region. For multi, both versions are acceptable. I've heard it both ways as well.(3 votes)

- how do i know that this is the 3d graph i get before starting to solve the problem?(2 votes)
- the partial derivative with respect to X equals 2XY

how come the graph is a shifted square At2:23?? shouldn't it be a straight line because Y is const?

and although the derivative equals a negative number (-2)

the point was in the positive Z axis

and at the same point the Y partial derivative gives a totally different value 1.9

then how did we draw this 3D graph?(1 vote)- You are correct, that the partial derivative with respect to x is 2xy, and that y is a constant. The plane (the cross section) of the graph at2:26is JUST how the function f, looks like, when y = +1, right? And we are saying, that indeed, the derivative of that parabolic looking cross section when y = 1 is 2xy. At2:47, that 2xy shows up. Right? So that parabola is just THE FUNCTION F, when y = 1, and NOT the derivative. When we take the derivative OF THAT CROSS SECTION, we get 2xy, just to emphasize. Have a nice day ); ^^(2 votes)

- how do you actually plot this two variable function and with respect to what are you finding the derivative..? it was pretty easy to visualize in a single variable...please do answer...thanx(1 vote)
- I have a doubt how x^2y+sin(y) is shown in 3d model it's a 2d right? that means function is z right! Why should it be taken like that? Since we have only 3d!(1 vote)

## Video transcript

- [Voiceover] Hello everyone. So I have here the graph
of a two-variable function and I'd like to talk about
how you can interpret the partial derivative of that function. So specifically, the function that you're looking at is f of x, y is equal to x squared times y plus sine of y. And the question is, if I
take the partial derivative of this function, so maybe I'm looking at the partial derivative
of f with respect to x, and let's say I want to do
this at negative one, one so I'll be looking at
the partial derivative at a specific point. How do you interpret
that on this whole graph? First, let's consider where
the point negative one, one is. If we're looking above, this is our x-axis, this is our y-axis the point negative one, one is sitting right there. So negative one, move up one and it's the point that's
sitting on the graph. And the first thing you might do is you say well, when we're
taking the partial derivative with respect to x, we're going to pretend that y is a constant so let's actually just go
ahead and evaluate that. When you're doing this, x squared looks like a variable, y looks like a constant, sine of y also looks like a constant. So this is going to be... We differentiate x squared and that's two times x times
y which is like a constant, and then the derivative of
a constant there is zero and we're evaluating this whole thing at x is equal to negative
one and y is equal to one. So when we actually plug that in, it would be two times negative
one multiplied by one, which is two... Negative two, excuse me. But what does that mean, right? We evaluate this, and
maybe you're thinking this is kind of slight
nudge in the x direction, this is the resulting nudge of f. What does that mean for the graph? Well first of all,
treating y as a constant is basically like slicing the whole graph with a plane that represents
a constant y value. So this is the y-axis, and the plane that cuts it perpendicularly that represents a constant y value. This one represents the
constant y value one but you could imagine sliding
the plane back and forth and that would represent
various different y values. So for the general partial derivative, you can imagine whichever one you want but this one is y equals one and I'll go ahead and slice
the actual graph at that point and draw a red line. And this red line is basically all the points on the graph
where y is equal to one. So I'll emphasize that...
where y is equal to one. This is y is equal to one. So, when we're looking at that we can actually interpret the
partial derivative as a slope because we're looking at the point here, we're asking how the function changes as we move in the x direction. From single variable calculus,
you might be familiar with thinking of that
as the slope of a line and to be a little more
concrete about this, I could say you're starting here, you consider some nudge over
there, just some tiny step. I'm drawing it as a sizable one but you imagine it as a
really small step, as your dx, and then the distance
to your function here the change in the value
of your function... I said dx, but I should say
partial x or del x... Partial f. And as that tiny nudge
gets smaller and smaller, this change here is going to correspond with what the tangent
line does, and that's why you have this rise over
run feeling for the slope. And you look at that
value, and the line itself looks like it has a slope
of about negative two so it should actually
make sense that we get negative two over here
given what we're looking at. But let's do this with the partial derivative
with respect to y. Let's erase what we've got going on here and I'll go ahead and move
the graph back to what it was, get rid of these guys, so now we're no longer
slicing with respect to y, but instead let's say we slice
it with a constant x value. So this here is the x-axis;
this plane represents the constant value x equals negative one and we can slice the graph there. Kind of slice it, I'll
draw the red line again that represents the curve and this time, that curve represents that value x equals negative one. It's all the points on the graph where x equals negative one. And now when we take
the partial derivative, we're going to interpret it as a slice... As the slope of this resulting curve. So that slope ends up looking like this, that's our blue line, and
let's go ahead and evaluate the partial derivative
of f with respect to y. So I'll go over here,
use a different color so the partial derivative of f
with respect to y, partial y. So we go up here, and it says,
okay. So x squared times y. It's considering x squared
to be a constant now. So it looks at that and
says x, you're a constant, y, you're the variable,
constant times a variable, the derivative is just
equal to that constant. So that x squared. And over here, sine of y, the derivative of that with respect to y is cosine y. Cosine y. And if we actually want to evaluate this at our point negative one, one what we'd get is negative one squared plus cosine of one. And I'm not sure what the cosine of one is but it's something a little bit positive, and the ultimate result that we see here is going to be one plus something, I don't know what it is,
but it's something positive, and that should make sense
'cause we look at the slope here and it's a little bit more than one. I'm not sure exactly, but it's
a little bit more than one. So you often hear about people talking about the partial derivative as being the slope of
the slice of a graph. Which is great, if you're looking at a function that has a two-variable input and
a one-variable output, so that we can think about its graph. And in other contexts,
that might not be the case. Maybe it's something that has
a multidimensional output, we'll talk about that later, when you have a vector-valued function, what its partial derivative looks like, but maybe it's also something
that has a hundred inputs and you certainly can't
visualize the graph but the general idea of saying, "Well, if you take a tiny
step in a direction"-- here, I'll actually walk through it in this graph's context again. You're looking at your point here and you say we're going
to take a tiny step in the y direction. And I'll call that partial y. And you say that makes
some kind of change, it causes a change in the function which you'll call partial f. And as you imagine this
getting really really small, and the resulting change
also getting really small the rise over run of
that is going to give you the slope of the tangent line. So this is just one way of
interpreting that ratio, the change in the output that corresponds to a
little nudge in the input. But later on we'll talk
about different ways that you can do that. So I think graphs are
very useful (laughs)... When I move that, the text doesn't move. I think graphs are very
useful for thinking about these things, but
they're not the only way and I don't want you to
get too attached to graphs even though they can be handy in the context of two-variable
input, one-variable output. See you next video!