# Partial derivatives and graphs

- [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
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!