Dy/dx as Fraction: A-Level Maths/Further Maths Explanation

[barnaby]

I started calculus in September (as part of A-Level Maths/Further Maths), and we've been told time after time not to look at derivatives as fractions.

We recently did the Chain Rule, and we were told that a way to remember it was that if dy/dx = dy/du * du/dx, then the 'du's 'cancel out' - which flies in the face of not looking at derivatives as fractions. My teacher then told us that we would eventually see how derivatives could be treated like fractions...

How and why can we do this?

 

[EnumaElish]

See this thread: https://www.physicsforums.com/showthread.php?t=203204

 

[arildno]

I started calculus in September (as part of A-Level Maths/Further Maths), and we've been told time after time not to look at derivatives as fractions.

Remember that a "fraction" is a relation between two NUMBERS.
Since you cannot regard "dx" and "dy" as numbers, it follows that the symbol dx/dy cannot be regarded as a fraction.

 

[phoenixthoth]

Remember that a "fraction" is a relation between two NUMBERS.
Since you cannot regard "dx" and "dy" as numbers, it follows that the symbol dx/dy cannot be regarded as a fraction.

Actually, dx and dy are infinitesimal numbers, so it follows that the symbol dy/dx can be regarded as a fraction. Furthermore, it is not a coincidence that the chain rule works as though something is being canceled...because something is being canceled.

See, for example, chapter 2 of
http://www.math.wisc.edu/~keisler/foundations.pdf

 

[mathwonk]

actually dy and dx are differentials, i.e. sections of a certain line bundle, hence functions whose values are elements of a one dimensional vector space. now as long as a vector w is non zero, the quotient v/w of two vectors in a one dimensional vector space IS a number. so fractions do make sense and have numbers as values, as long as the top and bottom of the fraction are vectors in the same "line".

there is actually a good, elementary explanation of differentials in the beginning of the classic diff eq book by tanenbaum and pollard.

 

[phoenixthoth]

actually dy and dx are differentials, i.e. sections of a certain line bundle, hence functions whose values are elements of a one dimensional vector space..

...that one dimensional space being the hyperreals, to be more specific, which is what I said.