Linked Questions

7
votes
3answers
3k views

Definition of e [duplicate]

Possible Duplicate: Why is $1^{\infty}$ considered to be an indeterminate form Is $dy/dx$ not a ratio? I'm very eager to know and understand the definition of $e$. Textbooks define $e$ as ...
1
vote
3answers
1k 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 ...
0
votes
2answers
109 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
108 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 ...
0
votes
0answers
46 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 ...
69
votes
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 ...
78
votes
20answers
9k 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 ...
50
votes
11answers
6k 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 ...
79
votes
3answers
37k 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 ...
39
votes
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 ...
18
votes
5answers
2k views

Chain Rule Intuition

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 ...
15
votes
6answers
2k 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 ...
14
votes
3answers
2k views

If $\frac{dy}{dt}dt$ doesn't cancel, then what do you call it?

I have $y$ is a function of $t$. I have reached a situation here where I need to evaluate $$\displaystyle \int_0^b{\frac{dy}{dt}dt}$$ Now clearly $y$ has dependence on $t$, otherwise $\displaystyle ...
17
votes
2answers
2k views

Physicists, not mathematicians, can multiply both sides with $dx$ - why?

The following question is asked without malicious intentions - it's not intended as a flamebait! In my physics textbooks (Young & Freedman in particular) I have often seen derivations of ...
16
votes
4answers
1k views

Can I ever go wrong if I keep thinking of derivatives as ratios?

I have been forewarned about it, I have read the answers here, but I haven't seen a counter example where it doesn't work. I know that it isnt really a fraction, but does it effectively get the same ...

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