Partial Derivatives (Part - 2) - Mathematics, Engineering Video Lecture - Engineering Mathematics | Best Video for Engineering Mathematics

Before we move on to other
functions, let's also take the
partial derivative of our
function here, f of xy, or the
partial derivative of
z with respect to y.
So let's do it in magenta.
So the partial derivative
of z with respect to y.
Well, now we're saying how much
does z change with respect
y if x is constant.
So this x squared we
treat as a constant now.
So the derivative of a
constant with respect to
y is 0, so ignore it.
Now this xy term.
The way we're doing it
now is, y is a variable,
x is a constant.
So what is the derivative
of, I don't know, 5y
with respect to y?
Well it's 5.
So the derivative of xy with
respect to y is just x.
And what's the derivative of
y squared with respect to y?
Well it's just 2y.
So you can see it's
quite symmetric.
The partial of z with respect
to x is 2x plus y, the
partial of z with respect
to y is x plus 2y.
That's because this equation
is pretty symmetric.
The x's and the y's kind
of do the same thing.
Now, we picked the point x is
equal to 0.2 y is equal to 0.3.
Actually let me erase this,
because I picked a different
point where I graph, and I
graphed it ahead of time
just to save time.
So I don't think I have
to include this anymore.
So what I did, I picked
the point x is equal to
0.3, y is equal to 0.3.
And when x is equal to
0.3, y is equal to 0.3,
what is z equal to?
Let's see, 0.3 squared is
0.09, 0.3 times 0.3 0.09.
So it's z is equal
to 0.27, right?
Just substitute 0.3
in for x and y.
z is equal to 0.27.
So what is the partial
of z with respect
to x at that point?
Or we could write, f
sub x at the point y.
x is equal to 0.3,
y is equal to 0.3.
It equals, we figured that,
let's see 2 times 0.3 is 0.6,
plus 0.3, that's equal to 0.9.
So the slope in the x direction
at that point is 0.9.
And if we take the partial with
respect to y at that same
point, 0.3 plus 0.6,
that's also equal to 0.9.
Let's see if we can
visualize this.
Let me bring in my graph.
There we go.
So this is, this surface once
again is the surface of z is
equal to x squared plus
xy plus y squared.
And this box is kind of the
domain, and the x and y
dimensions that I define.
Kind of, I bounded it because
it starts to increase really
fast, and you wouldn't be able
to see all this interesting
stuff that happens closer in.
But this, what I did, so this
vertical line, I just wanted
to show you that when x is
equal to 0.3, y is equal
to 0.3, z is equal to 0.7.
So that just kind of helps show
you that, ok, that shows what
point we're working with.
And then these two lines, this
is, if you think about it, this
is the line where y
is constant, right?
So this is the slope in the
z, or as the surface changes
with respect to x at
this point, right?
This is the tangent
line relative to x.
So you could kind of view
it as, if you hold y
constant, here's a tangent
line at that point.
And if you hold x constant,
here's a tangent
line at that point.
And like I said in the last
video, you can actually have
infinite tangent lines.
You have to pick the direction
that you want to go in the xy
plane, and then you could
plot a tangent line.
And so that's why we did
partial derivatives
to begin with.
Actually, this is pretty cool.
We can actually take, we can
actually zoom in on this.
Zoom it a little bit more.
I want to zoom in on the
part that is interesting,
let me translate this.
So that's the part
that's interesting.
And now let me rotate it.
So you can actually rotate.
So this is the tangent.
This shows that the partial
of the function with respect
to y, the slope is 0.9.
And this line shows that the
partial of the slope, or the
partial the function, sorry the
partial of z, or the partial of
the function with respect to
x is 0.9, at this point.
At the point x is 0.3, y is
0.3, right? x is 0.3, y is
0.3, z is equal to 0.27.
And we can rotate it just
to get more intuition.
I think it's, the graphing
thing looks a little
bit funny sometimes.
But you see that both
of those lines are
tangent at that point.
And in fact, two lines define a
plane, and the plane that's
defined by those two lines, or
any of the two tangent lines to
that point, defines a tangent
plane to the surface.
So a does have only one tangent
plane, but within a tangent
plane, there are an infinite
number of tangent lines.
Well anyway, that's the
fun with graphing.
Now let's just chug through a
bunch of partial derivative
problems just so that you get
used to the mathematics of it.
Delete that.
Let's do some that might
confuse you, so you
see how to do them.
Let's say that f of xy,
and I'm confining it
to three dimensions.
Although we can do it more.
Actually, maybe I'll do it more
dimensions now that we're not
going to try to visualize it.
Let's say it's x sine
of x, cosine of y.
So let's take the partial
of f with respect to x.
This is still going to be
a function of x and y.
So we treat y like a constant
So cosine of a constant, this
is just going to be a constant.
So we can almost ignore that.
We could put that out front.
We could say that it's going
to be cosine of y times the
derivative of this
with respect to x.
So you could say, cosine
of y, it's just a number.
This cosine of y could
just be, I don't know,
5 or pi, or whatever.
Cosine of y.
And then, because when you take
the derivative the constant
just comes out of
the derivative.
And then we would take the
derivative of the x's.
So the derivative of the
first term with respect
to x, well that's just 1.
Times the second expression.
So sine of x, I'm just doing
the product rule here.
Plus the derivative of
the second expression,
that's cosine of x.
Cosine of x is the derivative
of the second expression, times
the first expression, times x.
So If we wanted to expand it
all out, the partial of f
with respect to x, this is
the function of x and y.
It equals sine of x, cosine of
y, plus, let's put this x out
front, just so we change the
order. x, cosine of
x, cosine of y.
Not too difficult.
You just have to realize
that anything with
the y is a constant.
So let's reverse it.
Well, not reverse it.
Let's take the partial
now in the y direction.
How much does f change
in the y direction if
we hold x constant.
So the partial of f with
respect to y, still a
function of x and y.
The derivative in
that direction is a
function of x and y.
So now x is a constant.
So this actually becomes
pretty straightforward.
This whole x, sine of x,
if x is some number, 5,
this is just a constant.
So we can just write
that out front.
So that's just x, sine of x.
I know it's hard for you to get
used to saying that, oh, x sine
of x, that's just a
constant number.
Because you're so used to
taking the derivative
with respect to x.
And that's the hardest
part about doing these
partial derivatives.
But anyway, this is
just a constant term.
And now we just take
the derivative of this
with respect to y.
The derivative of cosine
of y with respect to
y is minus sine of y.
I'll do that in yellow.
Minus sine of y.
I just want to put
the minus out front.
Minus sine of y.
There you have it.
Let's do another one.
And actually I'm going
to add more variables.
Let's say that, I don't know,
just so you get used to
the different notation.
x is equal to a squared times,
I don't know, a squared times b
to the third, times
c to the 1/2 power.
Now what is the partial
derivative of x
with respect to a?
Well everything else
is just a constant.
What's the partial of a
squared with respect to a?
Well it's 2a.
So it'll just be 2a
times the constants.
Times b to the third,
c to the 1/2.
I can actually get rid of
my parentheses there.
What's the partial derivative
with respect to b?
Ah, sorry, what's the partial
of x with respect to b?
Well now, a squared and c to
the 1/2 are just constants.
We can just write that. a
squared, c to the 1/2.
And now we just take the
derivative with respect to b.
Well that's 3b squared.
Times 3b squared.
If I just want to rearrange
it, that's 3a squared, b
squared, c to the 1/2 power.
Not too difficult.
You just have to keep in mind
what's constant and what's not.
And then finally, the partial
of x with respect to c.
a squared, b to the third,
those are both constants.
a squared, b to the third
times derivative of
this with respect to c.
1/2 c to the minus 1/2.
Or we could rewrite this as
a squared, b to the third,
over 2 square root of c.
Just a little bit of
algebraic manipulation.
Anyway, I will see you
in the next video.