If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## Multivariable calculus

### Course: Multivariable calculus>Unit 2

Lesson 5: Multivariable chain rule

# 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? • 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
• 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? • 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.
• 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? • 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.
• 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? • 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? • 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. • 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)? • What if x and y are also multivariable functions ? • how about finding the multivariable chain rule without paramtrizing with the parameters t. Since Z=x²y itself is a function..  