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## Multivariable calculus

### Course: Multivariable calculus>Unit 2

Lesson 5: Multivariable chain rule

# Multivariable chain rule and directional derivatives

See how the multivariable chain rule can be expressed in terms of the directional derivative. Created by Grant Sanderson.

## Want to join the conversation?

• I don't think the interpretation was well organized enough. I got very confused. • I agree. The many rapid changes between forms of notation without worked-through examples led to a falling-off in the explanatory value of the video. I understand what Grant is getting at, but this video did not add to my understanding. It should concentrate either on explaining how the multivariable chain rule spits out the directional derivative or on showing how the rule can be expressed using different forms of notation, but not on both as this causes understanding of the relationship between the multivariable chain rule and the directional derivative to be lost.
• This quickly became very hard to follow. The incredibly compact expression is quite complex, and confusing. • I also have difficulty in getting the intuition, but I think what Grant is getting at is this:

Think of the derivative of f with respect to t as the rate of change for f as you change t by a little (in fact infinitesimal) bit. The directional derivative definition says that this is equal to the directional derivative of f with respect to v'(t), evaluated at v(t). The directional derivative itself is how much f changes as we go along in the v'(t) direction (the direction of motion). We evaluate this at v(t) because that is the point in which the change is occurring.

Hope this helps!
• I think it is great that sometimes things are taught in general terms, but most of the time you should be doing examples with real problems. Then everyone will get a great understanding. Obviously, there is a lot of confusion from other people here. Especially in more advanced math. Where are all my example problems and all the challenges that we had in all the lessons before? • Shouldn't the W vector in the directional derivative equation around be a unit vector? • I think the comments and criticisms that this video is confusing, badly organised and lacking examples are simply wrong in the context of all the prior videos in this series and what this video clearly set out to achieve. I suggest you look back at all the previous videos in the series where the 2-dimensional cases are clearly and painstakingly laid out and worked through. This video was clearly indicated as the step to generalise the previous examples and results, showing how the vector notation is a compact way to present results for functions of variables of any dimension. • I watched all preceding multivariable calculus videos, and they were easy to understand. However, this one is a bit too abstract. Maybe it would help to provide some example problems. • why are we not dividing by the modulus of w as was done in directional derivative
(1 vote) • Simply because we are not taking the slope of f with respect to a nudge in the direction of w in an xy plane, but the derivative of f with respect to t which causes a change in w. So the magnitude of w matters. Loosely speaking, if a change in t causes a large change in w, the derivative of f would be larger (simply because a larger change in w should cause a larger change in f) and if the change in w is small the change in f is small.
• Instead of clarifying what I saw in class, this video, confused me more
(1 vote) • v'(t) in this case is a unit vector, right?
(1 vote) • On the side, can I just verify something:

So a directional derivative should be with a normalized vector, but despite the similarity to chain-rule for vector derivatives, there's no requirement for normalization of the vector input into the function, correct?
(1 vote) 