Linked Questions
209 questions linked to/from Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
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Question in the definition of curvature
$$K = \Bigg|\frac{dT}{ds}\Bigg| = \Bigg|\frac{\frac{dT}{dt}}{\frac{ds}{dt}}\Bigg|$$
where $T$ is the unit tangent vector to the curve and $s$ is arc lenght.
I don't know why that relationship is True ...
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Is $\frac{dy}{dx}$ a ratio? [duplicate]
In differential calculus for beginners, Joseph Edwards alerts the reader against the fallacious notion of $\frac{dy}{dx}$ being a ratio.
But then in Integral calculus they play around with $\ dy$ and $...
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Why can we treat dy/dx as a fraction in 1 dimension [duplicate]
When doing integration and calculus why can we manipulate $dy/dx$ as if it was a fraction. I heard this only works when working in one dimension or something? And aren't $dy$ and $dx$ supposed to be ...
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The dy/dx notation [duplicate]
In the dx/dy notation, we could interpret it is a fraction, right? Thus, the Ds cancel out? So that would result in x/y? Am I right?
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When can we safely treat $dx$ like it were a real number during integration by substitution
$\def\R{\mathbf{R}}$
Disclaimer. I am aware that this is similar to this post, but the difference here is I'm asking, when can we safely treat these 'infinitesimal' quantities as though they were real ...
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Solving $y' = \frac{x^2 + 2xy - y^2 - 2}{x^2 - 2xy - y^2 + 2}$ without tricks
I would like to solve the equation:
$y' = \frac{y^2 - 2xy - x^2 + 2}{y^2 + 2xy - x^2 - 2} \tag{1}$
From that, we have:
$y'(y^2 + 2xy - x^2 - 2) = y^2 - 2xy - x^2 + 2 \implies $
$\frac{(2x + 2yy')(x+...
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Did Newton and Leibniz use limits in their derivations of differential calculus?
In modern treatments of calculus, limits are used to motivate the derivation of differential calculus. However, when I searched for the history of limits I came across this Wikipedia article, which ...
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Given an equation of differentials, how can I find the relative rate of change
Context
I am studying special relativity. In [1], Gray derives an equation that is written in terms of three differentials. Namely,
$$\left(d\tau\right)^2 = \left(dt\right)^2 - \frac{\left|d\mathbf{r}\...
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Difference between $dy$ and $dx$ [duplicate]
I've been taught about $dy/dx$ and how it can be split into $\frac{dy}{dx}=\frac{d}{dx}y$. I'm confused as to why this happens. Don't $dy$ and $dx$ both refer to infinitely small changes in their ...
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How can we treat dy and dx as quantities? [duplicate]
It is very common in non-rigorous calculus classes to perform operations of the form $$\frac{dy}{dx} = f'\left(x\right) \implies dy = f'\left(x\right) dx$$ As if these are simply numbers. How is this ...
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Show that the distribution defined by $f_n(x)=\sin(nx)$ converges to $T_0$
Let $n>0$ and define the function $(f_n)$ as $f_n(x)=\sin(nx)$.
Show that $T_{f_n} \underset{\mathcal{D'}}{\rightarrow} T_0$.
As a hint, I was told to use the fact that for distributions one can ...
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Method of differentiation
First of all we all know that d/dx is an operator which differentiates the function provided to it w.r.t. x okay!
Now, while differentiating complex functions we often use Chain Rule. Sometimes during ...
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How does the idea of a differential $\text{d}x$ work if derivatives are not fractions?
I was going through some old work and had a question about differentials.
I know that derivatives $\frac{\text{d}y(x)}{\text{d}x}$ are not fractions, but rather an operator $\frac{\text{d}}{\text{d}{x}...
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Why do we use the $\mathrm dy/\mathrm dx$ definition when talking about rates and not $\Delta y/\Delta x$
I am a student learning rates of change. Why is it always that people use the $\,dy/\,dx$ to represent a rate of change and not the definition of the slope which is $\Delta y/ \Delta x$
I think using $...
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question about flipping chain rule derivatives
If $$\frac{dy}{dx} = \left(\frac{dy}{du}\right)\left(\frac{du}{dx}\right),$$ will the inverse of $dy/dx$ flip the other derivatives? For instance, will $$\frac{dx}{dy} = \left(\frac{du}{dy}\right)\...
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