Hey everyone!
Welcome back!
Today we’re going to be doing some Partial Derivative problems, the first one that we’re
going to do is f of (x, y, z) equals x squared times y cubed times z to the four, and all
that partial derivatives means is that we’re going to be taking the derivative for every
single variable in the problem.
So, since there are three variables, we’re actually going to have to take the derivative
three times, once for each variable, and its’ going to be a separate equation four each
one.
So we’ll go in order.
The first one we’ll look at is x and the notation for partial derivative looks like
this.
This weird squiggly thing f, the weird squiggly thing x, so f, you get from here, so this
is, you know, h of (x, y, z) then this becomes an h and then x is the first variable we’re
going to… we’re going to do.
So, we’ll go ahead and say…
now, when we’re taking and… and when…
when we take the partial derivative here with x, we say that we’re taking the partial
derivative with respect to x, that’s how people call it, then we’ll take the partial
derivative with respect to y then with respect to z.
So, looking at x, the way that we take the partial derivative with respect to x while
still having these other variables in the equation is we treat the other variables like
they’re constants.
And, what I like to do, and you get…
you’ll get faster and faster at it in your head, but the way that I like to do that just
to make it really obvious because sometimes it’s hard to understand how to hold those
constant especially when…
when you’re first starting out, I like to actually put a constant in there for those
numbers and then simplify the equation and then take the partial derivative so I can
see it.
So… so, what I would do here, for example, we’re talking about holding y and z constant,
as if they we’re a constant number like two or three, so let’s go ahead and put
two in for… for… for y and z here.
If we did, we would have… we would have x squared times two cubed times two to the
fourth, right, because we…
we plugged in two for y and for z.
Okay?
So, since we’re keeping this constant, this is how the equation would simplify.
So if… if you… if you multiply this out, let’s see, this would actually be x squared
times eight and this would be times sixteen, so this would be eighty and forty eight, so
this would be x squared times a hundred and twenty eight, right, if you simplify that.
So, what would be if we…
if we were taking the derivative of this normally, we would be looking at a hundred and twenty
eight x squared.
We would take the derivative of this and if we would multiply two times the coefficient
which is a hundred and twenty eight so that would be, what?
Two hundred and fifty six, so the derivative of this would be two hundred and fifty six
x, right?
So, what I’m hoping that you can see from this is that it’s… its... it’s exactly
the same thing.
We’re going to hold these two things constant and they are going to be like a coefficient
and… and this two hundred and fifty six stays.
So, this is actually going to be an…
Let’s go ahead and write out the answer and then we’ll compare them.
You’re going to multiply these two out in front here so it’s going to be two x and
then y cubed z to the four.
That’s going to be the answer for the partial derivative and I that you can see the relationship
here.
We multiplied the two on the x squared out in front just like we did here, we…
we brought this two out in front, and we ended up with a single x just like we ended up with
a single x here and we left y cubed and z to the fourth because they were absorbed into
the coefficient here.
They are like… because they’re constants and they’re multiplied together, they are
part of the coefficient, they’re like part of this two which is why… which is why they
get left in… in this equation.
Let’s go ahead and do y so that we see in another example and hopefully we’ll start
to understand.
So, when we…
when we take the partial derivative with respect to y as you might expect, it’s going to
be a partial derivative of f with respect to y, just like we did for x here.
So now, with y, we’re going to actually be holding…
Oh, I hope you guys can’t hear that fire truck.
So with… with y, we’re going to be keeping x and z constant so they’re going to be
like the coefficient as well, you could plug in numbers for them and… and go through
the same exercise.
But, they’re like the coefficient so they’re going to stay exactly the same because they’re…
they’re multiplied here with the y so we’re not even going to touch them.
Remember, we didn’t touched y cubed, we didn’t touch z to the fourth so x and z,
this time, are going to stay as well.
So, all we’re really looking at is… is the y and we’re going to… we’re going
to do the same thing we did with x, take the derivative of y.
So we’re going to get that three out in front and then y squared, right, three y squared
is the derivative of y cubed.
So we took the derivative and then the x squared and the z to the fourth are just going to
stay.
So, that’s the derivative with respect to y.
I left the space because when you take the… the partial derivative, you always like to
keep the variables in alphabetical order.
So, I could have written three y squared x squared z to the fourth but we like to always
keep them x, y, z in order.
So, we’ll go ahead and… and do the same thing here for z.
So it’s going to be the partial derivative of f with respect to z and we will go ahead
and leave x squared and y cubed.
We’re not touching them because they’re like part of the coefficient, they stay.
So we’ll go ahead and say x squared y cubed and then we…
we take the derivative of z here.
So the derivative…
Of course, we subtract one from the exponent, so four minus one is three and the four gets
multiply out in front so it comes out here.
So our answer with respect to z is actually four x squared y cubed z cubed.
And, your final answer is… is a three part answer if you’re asked to take the… the…
the derivative of this function or the partial derivative.
Because there are three variables, you need each one of these equations and you would
want to write all three of these down on the homework or on your test because… because
your answer is actually all three of these.
So, there you have it.