Linked Questions

2
votes
3answers
5k 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
3answers
645 views

If $F=m\dfrac{dv}{dt}$ why is it incorrect to write $F\,dt=m\,dv$? [duplicate]

My university lecturer told me that: If $$F=m\dfrac{dv}{dt}$$ it's incorrect to write $$F\,dt=m\,dv\tag{1}$$ but it is okay to write $$\int F\,dt=\int m\,dv$$ ...
3
votes
2answers
868 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 ...
6
votes
2answers
717 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
467 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. ...
0
votes
1answer
470 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 ...
2
votes
3answers
234 views

Rigorous definition of differentials in the context of integrals. [duplicate]

When using the subsitituion rule in integration of an integral $\displaystyle \int f(x)\,\mathrm dx$, one turns the integral from the form $$\displaystyle \int f(x(t))\,\mathrm dx \quad(*)$$ into the ...
5
votes
2answers
248 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(...
6
votes
1answer
262 views

Why isn't it mathematically rigorous to treat dx's and dy's as variables? [duplicate]

If I do something like: $$\frac{dy}{dx} = D$$ $$dy = D \times dx$$ People would often say that it is not rigorous to do so. But if we start from the definition of the derivative: $$\lim_{h \to ...
0
votes
1answer
298 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
165 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 ...
2
votes
2answers
71 views

Handling $\frac{dy}{dx}$ as a ratio. [duplicate]

I know that given a differential equation, one that is separable, it is not fully correct to handle the $\frac{dy}{dx}$ as a ratio. Meaning that it is not simply a small difference in $y$ over a small ...
0
votes
1answer
127 views

Derivatives interpreted as fractions [duplicate]

So I was wondering if manipulating differential as fraction is fine.In Physics problems ,usually, what I see is if we have equations of magnetic field ,we consider a magnetic field caused by a small ...
1
vote
1answer
74 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 ...
2
votes
0answers
111 views

why do we use dy/dx as ratio though it is not while solving the problems of integration by substitution [duplicate]

According to my knowledge dy/dx is not a ratio. Then while solving the problems of integration by substitution how can we use it as ratio. Because of we have dx/dt =f(x). Then while shoving it by ...

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