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Current time:0:00Total duration:6:55

- [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!