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### 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 ...
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### $dy\over dx$ 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 ...
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### 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 ...
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 $dy/dx = ... 1answer 46 views ### Why do we treat differential notation as a fraction in u-substitution method [duplicate] How did we come to know that treating the differential notation as a fraction will help us in finding the integral. And how do we know about its validity? How can$\frac{dy}{dx}$be treated as a ... 4answers 92 views ###$dx=\frac {dx}{dt}dt $. Why is this equality true and what does it mean? [duplicate]$dx=\frac {dx}{dt}dt $. I know that this deduction is obvious from the chain rule, given that we treat our dx and dt as just numbers. But I find it quite unsatisfactory to think of it in that sense. ... 0answers 48 views ### Why is$\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$, but when it comes to implicit differentiation and integration ... 0answers 18 views ### flexibility of differention operator dy/dx [duplicate] If dy/dx = p(say) then we write dy = pdx .But in reality d /dx is an operator,not simply division.So we can not treat this operator as simple division but we do.In reality we can not seperate this dy ... 31answers 7k views ### What are some conceptualizations that work in mathematics but are not strictly true? I'm having an argument with someone who thinks it's never justified to teach something that's not strictly correct. I disagree: often, the pedagogically most efficient way to make progress is to ... 22answers 10k views ### Most ambiguous and inconsistent phrases and notations in maths What are some examples of notations and words in maths which have been overused or abused to the point of them being almost completely ambiguous when presented in new contexts? For instance, a ... 11answers 7k views ### What is$dx$in integration? When I was at school and learning integration in maths class at A Level my teacher wrote things like this on the board. $$\int f(x)\, dx$$ When he came to explain the meaning of the$dx$, he told us ... 3answers 44k views ### What is the practical difference between a differential and a derivative? I ask because, as a first-year calculus student, I am running into the fact that I didn't quite get this down when understanding the derivative: So, a derivative is the rate of change of a function ... 9answers 2k views ### Is$dx\,dy$really a multiplication of$dx$and$dy$? On the answers of the question Is$\frac{dy}{dx}$not a ratio? it was told that$\frac{dy}{dx}$cannot be seen as a quotient, even though it looks like a fraction. My question is: does$dxdy$in the ... 7answers 3k views ### What does$dx$mean?$dx$appears in differential equations, such us derivatives and integrals. For example, a function$f(x)$its first derivative is$\dfrac{d}{dx}f(x)$and its integral$\displaystyle\int f(x)dx\$. But ...
We know that the chain rule is used to differentiate a composite function ,say $$f(x) = h(g(x))$$ It's defined as the derivative of the outside function times the derivative of the inner function or ...