2
votes
3answers
3k views

What is wrong with treating $\dfrac {dy}{dx}$ as a fraction? [duplicate]

If you think about the limit definition of the derivative, $dy$ represents $$\lim_{h\rightarrow 0}\dfrac {f(x+h)-f(x)}{h}$$, and $dx$ represents $$\lim_{h\rightarrow 0}$$ . So you have a $\;\;$$\...
5
votes
2answers
273 views

How is an infinitesimal $dx$ different from $\Delta x\,$? [duplicate]

When I learned calc, I was always taught $$\frac{df}{dx}= f'(x) = \lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{(x+h)-x}$$ But I have heard $dx$ is called an infinitesimal and I don't know what this means....
3
votes
3answers
191 views

$dx=\frac {dx}{dt}dt $. Why is this equality true and what does it mean? [duplicate]

$dx=\frac {dx}{dt}dt $. I know that this deduction is obvious from the chain rule, given that we treat our dx and dt as just numbers. But I find it quite unsatisfactory to think of it in that sense. ...
3
votes
2answers
102 views

Differential Notation Magic in Integration by u-Substitution [duplicate]

I'm really confused now. I always thought that the differential notation $\frac{df}{dx}$ was just that, a notation. But somehow when doing integration by u-substitution I'm told that you can turn ...
-1
votes
1answer
149 views

Why do we treat differential notation as a fraction in u-substitution method [duplicate]

How did we come to know that treating the differential notation as a fraction will help us in finding the integral. And how do we know about its validity? How can $\frac{dy}{dx}$ be treated as a ...
5
votes
2answers
90 views

Conceptual question on substitution in integration [duplicate]

In calculus we learn about the substitution method of integrals, but I haven't been able to prove that it works. I mainly don't see how manipulations of differentials is justified, i.e how $dy/dx = f(...
0
votes
1answer
156 views

$dy\over dx$ is one things but why in integration we can treat it as 2 different terms [duplicate]

when i am learning differentiation, my lectuer tell us that the deriative $dy\over dx$ is one things, it is not the ration between dy and dx. However when i learn about integrating, sometime we need ...
1
vote
0answers
141 views

How do we go from $f'(x) = \frac{dy}{dx}$ to $dy = f'(x)dx$? [duplicate]

As far as I know, the derivative of $y$ is defined as: $$f'(x) = \lim_{h \rightarrow 0} \frac{f(x + h) - f(x)}h = \frac{dy}{dx}$$ So $\frac{dy}{dx}$ is a limit, not a fraction of real numbers. I ...
1
vote
1answer
47 views

Intuition behind calculus notation [duplicate]

I have read somewhere sometime ago (very specific, I know...), that it would not be correct to treat rate of change, i.e. any arbitrary $\frac{dy}{dx} \ $ as a fraction, and thus it is not possible to ...
0
votes
0answers
59 views

Why is $\frac {dy}{dx}$ treated as a fraction? Plus an implicit differentiation question. [duplicate]

Why is $\frac {dy}{dx}$ treated as a fraction? I always thought that it is just notation for the derivative of $y$ with respect to $x$, but when it comes to implicit differentiation and integration ...
0
votes
0answers
54 views

Can and when are we able to treat $\frac{dy}{dx}$ as a fraction [duplicate]

this is something I have been wondering about for a while. I was never truly sure about if we could treat $\frac{dy}{dx}$ as a fraction or not. But I never had an issue with it, but then I began to ...
0
votes
0answers
18 views

When are we allowed to treat differential operators as 'fractions' [duplicate]

When is it justified to write: $$\frac{dy}{dx} =\frac{dy}{dt}\cdot\frac{dt}{dx} = \dot{y}/\dot{x}$$ When are we allowed to see that $\frac{dt}{dx} = \frac{1}{\frac{dx}{dt}}=\frac{1}{\dot{x}}$ I don'...
0
votes
0answers
9 views

Link to question regarding treating differential operator as a ratio [duplicate]

I have attempted to find the post which provides an explanation as to the circumstances in which we can treat $\frac{dy}{dx}$ as a ratio which appears to be used in solving separable DE's, but I have ...
107
votes
22answers
11k views

Most ambiguous and inconsistent phrases and notations in maths

What are some examples of notations and words in maths which have been overused or abused to the point of them being almost completely ambiguous when presented in new contexts? For instance, a ...
77
votes
32answers
8k views

What are some conceptualizations that work in mathematics but are not strictly true?

I'm having an argument with someone who thinks it's never justified to teach something that's not strictly correct. I disagree: often, the pedagogically most efficient way to make progress is to ...

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