Main content

### Course: Multivariable calculus > Unit 2

Lesson 5: Multivariable chain rule- Multivariable chain rule
- Multivariable chain rule intro
- Multivariable chain rule intuition
- Multivariable chain rule
- Vector form of the multivariable chain rule
- Multivariable chain rule and directional derivatives
- More formal treatment of multivariable chain rule

© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Multivariable chain rule intuition

Get a feel for what the multivariable is really saying, and how thinking about various "nudges" in space makes it intuitive. Created by Grant Sanderson.

## Want to join the conversation?

- How wrong is it to view dy/dx as a fraction? Does dy=2 dx mean the same as dy/dx =2?(5 votes)
- Considering how far blindly viewing a differential as a fraction has taken us, I dont think it's 'wrong' in any of the cases we learnt upto now, but only a less rigorous and amaeturish method for finding a proof.

If you've done Physics, you'll realise it's somewhat similar to those Energy-based derivations of equations (Those ones in which E_before = E_after). Although it isn't wrong to use the Law of Conservation of energy as the main argument, it's not rigorous enough and a bit of a cheat and I'm guessing it's a similar case here(16 votes)

- But what if df=df/dx*dx/dt*dt multiplied by df/dy*dy/dt*dt? Here both x and y affect df, so have do you know you add them?(8 votes)
- Even in this video, x and y both affect f. But if you notice, f is a single variable output. This implies that the output space is a number "line" and not a plane. Essentially and change in x produces a certain change on the number line and a change in y produces another change on the number line. So the total change in the output space is given by the addition of the individual changes by x and y respectively. The total magnitude of change would be a summation of the ratio of change (doh F by doh x) times the actual change in x and similarly for y.(12 votes)

- Why is the change in z given by 'adding' the change in x and change in y? Yes, change in y and change in z is responsible for a change in Z, but why is this statement expressed as a simple addition?(7 votes)
- It is a vectorial notation. Since the direction is already implied by i and j notation, we can simply "add" both the magnitude and the direction to get z. It's really just like saying 3 steps to the right and 4 steps ahead are the same thing as 5 steps diagonally(of the 3-4-5 right trangle), assuming our definition of the directions "right" and "ahead" are perpendicular to and therefore do not interfere with each other.(8 votes)

- What if f(x,y) resulted in a vector? Would we split f into two independent functions for each component and then evaluate like normal calculus?(2 votes)
- You mean if the original function was a vector function? Then yes, you'd split it into two functions first. Then, whether you apply the regular chain rule or the multivariable one depends on your variables. AFter differentiating, you'd put them back into their vector form.(3 votes)

- There is still one thing which intuitively doesn't make sense to me.

This splitting into a sum of two components dx and dy makes sense if we are talking about vectors, but in our case the thing we want to get is a scalar, the magnitude of change. If we think of it as a length of vector we should take into account the triangle inequality theorem that states that the sum of lengths of any two sides of the triangle is always greater than the length of the third side. Therefore our estimation of the change should be always greater than the actual change.

I guess I am missing something, but I'm not sure what.(2 votes) - Still not sure on this.

What does he mean by all these nudges?

dy = (dy / dt) dt

How do nudges relate to this equation? Conceptually (no need for long derivations), what does the right hand side of this equation mean?(2 votes) - Let
**r**(t)=x(t)**i**+y(t)**j**. The can we say that d/dt[f(**r**(t))] is the directional derivative of f(**r**(t)) in the direction of**r**’(t)?(2 votes) - What if x and y are also multivariable functions ?(2 votes)
- If they are single variable then you end up with one single differential, if they are multivariable then you end up with the components of partial derivatives.(1 vote)

- how about finding the multivariable chain rule without paramtrizing with the parameters t. Since Z=x²y itself is a function..(2 votes)
- The intuition behind df being affected by a combination of the changes due to dx and that due to dy is very nice. At6:30, what is the intuition to determine that df is simply the sum of these two components? How do we know that these two factors do not combine in some other way to affect df? How do we know, for example, that df is not the product of these two, or perhaps the change caused by dx plus twice the change caused by dy, or some other more complicated function involving both of the changes?(1 vote)
- Because the small change in f is a vector, and can be written as the sum of a small change in the x-direction plus a small change in the y-direction. Each term in the multivariable chain rule is one of these single-coordinate vectors.(2 votes)

## Video transcript

- [Voiceover] So in the last
video, I introduced this multi-variable chain
rule and here I want to explain a loose intuition
for why it's true, why you would expect
something like this to happen. So the way you think about
an expression like this, you have this multi-variable function f of xy and you're plugging things in, but just that function itself, you'll be thinking of taking a two dimensional space you know here's our xy plane, and then mapping it to, you
know, just a real number line and I'll think of this
as f, as the output. So somehow our whole
function takes things from this two dimensional space
and plugs it onto this output. T you're thinking of just another number line up here, so t, and then you've got separate functions here, you know x of t and y of t. X of t and y of t. Each of which take that same value for a specific input, you know it's not that they're acting on different inputs, x of some other input t
and y of some other input, it's the same one and then they move that somewhere to this output space which itself get's moved over. And in this way you're thinking of it as just a single variable function that goes from t and ultimately outputs f it's just that there's
multi-dimensional stuff happening in between and now if we start thinking about the derivative of it - what does that mean, what
does that mean for the conception of the picture
that we have going on here? Well, that bottom part, that dt you're thinking of as a
tiny change to t, right? So you're thinking of
it as kind of a nudge, I'll draw it as a sizable line here for like moving from
some original input over, but you might in
principal think of it as a very, very tiny nudge in t. And over here you'd say
well, that's gonna move your intermediary output in the xy plane to, you know maybe it'll
move it in some amount, again imagine this is a very small nudge, I'm going to give it some size here just so I can write into it and then whatever that nudge
in the output space right, it's a nudge in some direction that's going to correspond
to some change in f. Some change based on the
differential properties of the multi-variable function itself. And if we think about this, this change you might break it into components and say this shift here
has some kind of dx, some kind of shift in the x direction and some kind of dy, some
shift in the y direction. But you can actually reason
about what these should be coz it's not just an arbitrary change in x or an arbitrary change in y, it's the one that was caused by dt. So if I go over here, I might say that dx is caused by that dt and the whole meaning of the derivative, the whole meaning of the single variable derivative would be that when we take dx dt, this is the factor that
tells us, you know, a tiny nudge in t, how
much does that change the x component and if you want you could think of this as kind of
cancelling out the dts and you're just left with
x, but really you're saying there's a tiny nudge in
t and that results in a change in x and this derivative is what tells you the ratio between those sizes. And similarly, that change in y here, that change in y is gonna be somehow proportional to the change in t and that proportion is given by the derivative of y with respect to t that's the whole point of the derivative, no no, with respect to t and again you can kind of think of it as if you're cancelling out the ts and this is why the fractional writing, this Leibniz notation is
actually pretty helpful. You know, people will say,
oh mathematicians would like, share their heads at the idea of treating these like
fractions, but not only is it a useful thing to do coz it is a helpful mnemonic, it's
reflective of what you're gonna do when you make
a very formal argument. And I think I'll do that in
one of the following videos, I'll describe this in a
very, a much more formal way that's a little bit more airtight than the kind of hand-waving nudging around. But the intuition you
get from just writing this is a fraction is
basically the scaffolding for that formal argument, so it's a fine thing to do, I don't
think mathematicians are shaking their heads every time that a student or a teacher does this. But anyway, so this is kind of gives you what that dx is, what that dy is and then over here if you're saying how much does that change
the ultimate output of the f? You could say, well, your
nudge of size dx over here, you're wondering how much
that changes the output of f, that's the meaning of the
partial derivative, right. If we say we have the partial derivative with respect to x, what that means, is that if you take a tiny nudge of size x this is giving you the ratio between that and the ultimate change to
the output that you want. You could think of it like this partial x is cancelling out with
that dx if you wanted or you could just say,
this is a tiny nudge in x, this is going to result
in some change in f - I'm not sure what - but the meaning of the derivative is the
ratio between those two and that's what lets you figure it out. And similarly, you might
call this the change in f caused by x, like, due to x. Due to, I should say to dx. But that's not the only thing
changing the value of f right? That's not the only change happening in the input space, you also
have another change in f and this one I might say is due to dy. Due to that tiny shift in
y and what that's gonna be we know it's going to
be proportional to that tiny shift in y and the
proportionality constant - this is the meaning of
the partial derivative, that when you nudge y in some way it results in some kind of
nudge in f and the ratio between those two is what
the derivative gives. So ultimately, if you
put this all together what you'd say is there's
two different things causing an ultimate change to f. So if you put these together, and you want to know what the
total change in f is - so I might go over here and say the total change in f,
one of them is caused by partial f, partial x
- and I can multiply it by dx here, but really, we know that dx, the change there was in turn caused by dt so that in turn is caused by the change in the x component that was due to dt. That was of course of size dt. And then for similar
reasons, the other way that this changes in the y direction is a partial of f with respect to y but what caused that initial shift in y, you'd say that was a shift
in y that was due to t, and that size is dy dt times
dt, you could think of it. So slight nudge in t causes a change in y, that change in y causes the change in f and when you add those two together that's everything that's going
on, that's everything that influences the ultimate change in f. So then if you take this whole expression and you divide everything out by dt so you know, kind of
erase it from this side and put it over here, dt, this is your multi-variable chain rule, and of course I've just
written the same thing again but hopefully this gives
a little bit on intuition for how you're composing different nudges and why you wanna think about it that way. Of course, you can see this, and you see the partial f kind of
cancels out with that dx and this partial y kind of
cancels out with that dy and you're left with
the two different things that constitute a change in x, you know this one is only
partially the change in f, this is also partially the change in f, but together they give
the ultimate change in f and I think that gives
a very strong reason, if you break it down like
that, why this should be true.