A common question on mathematics is "why can't derivatives be treated as fractions?" I like the explanation that derivatives are not fractions, they are the limits of fractions, which are not the same type of mathematical object, so it is wrong to assume they behave in the same way. The answers I've seen give examples where treating derivatives as fractions yields correct solutions and other cases where it does not, but they do not proceed to show how to tackle the problems of the latter case in the correct way.
My question is: what is the correct way to solve calculus problems without treating derivatives as fractions?
I expect the answer to be quite complex, as otherwise people wouldn't use treat derivatives as fractions in the first place, they would do the problems properly. If possible, I would appreciate a simplified explanation (at the level of an A-level student) to preface the full explanation which could be useful to more advanced users.
Answer
This is really an issue of how derivatives and differentials interact. The relationship between the $3$ quantities $dx$, $dy$, and $dy/dx$ is
$$dy = \frac{dy}{dx}\; dx.$$
So if you have a differential equation
$$\frac{dy}{dx} = g(x,y)$$
you can multiply both sides by $dx$
$$\frac{dy}{dx}\; dx = g(x,y)\; dx$$
and then replace the left side by the relationship given above
$$dy = g(x,y)\; dx.$$
This looks like we've "treated $dy/dx$ as a fraction" because it looks like the $dx$'s were canceled.
No comments:
Post a Comment