Demystifying the Chain Rule in Calculus - Comments

[PeroK]

Greg Bernhardt submitted a new PF Insights post

Demystifying the Chain Rule in Calculus

Continue reading the Original PF Insights Post.

 

[Greg Bernhardt]

Congrats on your first Insight @PeroK!

 

[Mark44]

Nice article, @PeroK!

 

[Delta2]

Great insight, it addresses the main issues an average student (and i myself had) might stumble into when coming in first contact with the chain rule.

 

[lekh2003]

This is a very insightful insight. I have just begun studying calculus and I was extremely worried about this magical chain rule. This article was helpful in demystifying whatever I could understand from the insight .

 

[Tom.G]

The light type used, 33% saturation, makes it difficult to read. Any chance of increasing the amount of "ink" used? This thread, as all others, uses 50-75% saturation. And the reply box I'm typing in uses 98% saturation.
Thanks.

 

[JuanC97]

Thank you PeroK, I've found myself lost with things like this a couple of times and I agree when you say that the differential notation lacks of many things. The last issue that I lately found confusing was that you pointed out in equation (8):

Having a function f(g(x,t),x) write the partial derivative of f wrt x without having written the same term in both sides of the equation.

One option would be , being f_i the partial derivative of f wrt its i-th argument, write f_x = f₁(x,t) g_x+f₂ ,
this way you could avoid writting f₂ again as f_x and it would be the same as you suggested there (1,2 instead of X,Y).
Another option: Using differential notation you would have to use parenthesis and write explicitly that
f₁ = (∂f / ∂g) keeping the 2nd argument of f fixed, but that would bring notational clustering so better stick with the first option. :P

Fortunately, some people will read this article and they won't have to question all their knowwledge again as I did in that moment.

 

[lavinia]

Nice article.

I would only comment that in multivariate calculus one inevitably gets into directional derivatives. To understand these I think it is helpful to think of the derivative(or differential) of a function as a linear map on direction vectors. The Chain Rule then says that if you compose two functions, the derivative of the composition is the composition of the derivatives. In classical multivariable calculus this means you matrix multiply the Jacobian matrices.

Also thinking of the derivative in this way gives a conceptual framework for the Chain Rule.