Answer by NinjaDarth for Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
You could probably give it meaning as ratios. Starting from first principles, if $y = f(x)$, then you would write:$$dy = f'(x) dx,\\d^2y = f''(x) dx^2 + f'(x) d^2x,\\d^3y = f'''(x) dx^3 + 3 f''(x) d^2x...
View ArticleAnswer by Rudra for Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
It is incorrect to call it a ratio but it's heuristically handy to visualize it as a ratio where the $\frac{l}{\delta} $ talks about the times a certain $l>0$ can be divided by saying an exceedingly...
View ArticleAnswer by Bible Bot for Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
While it can be approximated by a ratio, the derivative itself is actuality the limit of a ratio. This is made more apparent when one moves to partial derivatives and $\frac{1}{\partial y/ \partial...
View ArticleAnswer by Lalit Tolani for Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
If I give my answer from an eye of physicist ,then you may think of following-For a particle moving along $x$-axis with variable velocity, We define instantaneous velocity $v$ of an object as rate of...
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Answer by Joe for Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
$dy/dx$ is quite possibly the most versatile piece of notation in mathematics. It can be interpreted asA shorthand for the limit of a quotient: $$ \frac{dy}{dx}=\lim_{\Delta x\to 0}\frac{\Delta...
View ArticleAnswer by zkutch for Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
Of course it is a fraction in appropriate definition.Let me add my view of answer, which is updated text from some other question.Accordingly, for example, Murray H. Protter, Charles B. Jr. Morrey -...
View ArticleAnswer by Jordan for Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
In a certain context, $\frac{dy}{dx}$ is a ratio.$\frac{dy}{dx}=s$ means:Standard calculus$\forall \epsilon\ \exists \delta\ \forall dx$If $0<|dx|\leq\delta$If $(x, y) = (x_0, y_0)$, but $(x, y)$...
View ArticleAnswer by user681293 for Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
The best way to understand $d$ is that of being an operator, with a simple rule$$df(x)=f'(x)dx$$If you take this definition then $dy/dx$ is indeed a ratio as it is stripping $f'(x)dx$ of...
View ArticleAnswer by Gustav Streicher for Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
The derivate $\frac{dy}{dx}$ is not a ratio, but rather a representation of a ratio within a limit.Similarly, $dx$ is a representation of $\Delta x$inside a limit with interaction. This interaction can...
View ArticleAnswer by Toby Bartels for Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
There are many answers here, but the simplest seems to be missing. So here it is:Yes, it is a ratio, for exactly the reason that you said in your question.
View ArticleAnswer by Dávid Kertész for Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
I am going to join @Jesse Madnick here, and try to interpret $\frac{dy}{dx}$ as a ratio. The idea is: lets interpret $dx$ and $dy$ as functions on $T\mathbb R^2$, as if they were differential forms....
View ArticleAnswer by Hawthorne for Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
To ask "Is $\frac{dy}{dx}$ a ratio or isn't it?" is like asking "Is $\sqrt 2$ a number or isn't it?" The answer depends on what you mean by "number". $\sqrt 2$ is not an Integer or a Rational number,...
View ArticleAnswer by kozenko for Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
Anything that can be said in mathematics can be said in at least 3 different ways...all things about derivation/derivatives depend on the meaning that is attached to the word: TANGENT.It is agreed that...
View ArticleAnswer by John Robertson for Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
Of course it is a ratio. $dy$ and $dx$ are differentials. Thus they act on tangent vectors, not on points. That is, they are functions on the tangent manifold that are linear on each fiber. On the...
View ArticleAnswer by Squirtle for Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
I realize this is an old post, but I think it's worth while to point out that in the so-called Quantum Calculus $\frac{dy}{dx}$ $is$ a ratio. The subject $starts$ off immediately by saying this is a...
View ArticleAnswer by Yiorgos S. Smyrlis for Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a...
$\boldsymbol{\dfrac{dy}{dx}}$ is definitely not a ratio - it is the limit (if it exists) of a ratio. This is Leibniz's notation of the derivative (c. 1670) which prevailed to the one of Newton...
View ArticleAnswer by Christopher King for Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
In most formulations, $\frac{dx}{dy}$ can not be interpreted as a ratio, as $dx$ and $dy$ do not actually exist in them. An exception to this is shown in this book. How it works, as Arturo said, is we...
View ArticleAnswer by Mikhail Katz for Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
In Leibniz's mathematics, if $y=x^2$ then $\frac{dy}{dx}$ would be "equal" to $2x$, but the meaning of "equality" to Leibniz was not the same as it is to us. He emphasized repeatedly (for example in...
View ArticleAnswer by jacques sassoon for Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
Assuming you're happy with $dy/dx$, when it becomes $\ldots dy$ and $\ldots dx$ it means that it follows that what precedes $dy$ in terms of $y$ is equal to what precedes $dx$ in terms of $x$."in terms...
View ArticleAnswer by asmeurer for Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
My favorite "counterexample" to the derivative acting like a ratio: the implicit differentiation formula for two variables. We have $$\frac{dy}{dx} = -\frac{\partial F/\partial x}{\partial F/\partial...
View ArticleAnswer by Brendan Cordy for Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
Typically, the $\frac{dy}{dx}$ notation is used to denote the derivative, which is defined as the limit we all know and love (see Arturo Magidin's answer). However, when working with differentials, one...
View ArticleAnswer by GdS for Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
$\frac{dy}{dx}$ is not a ratio - it is a symbol used to represent a limit.
View ArticleAnswer by Jesse Madnick for Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
Just to add some variety to the list of answers, I'm going to go against the grain here and say that you can, in an albeit silly way, interpret $dy/dx$ as a ratio of real numbers.For every...
View ArticleAnswer by Tobin Fricke for Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
It is best to think of $\frac{d}{dx}$ as an operator which takes the derivative, with respect to $x$, of whatever expression follows.
View ArticleAnswer by Mariano Suárez-Álvarez for Is $\frac{\textrm{d}y}{\textrm{d}x}$ not...
It is not a ratio, just as $dx$ is not a product.
View ArticleAnswer by Arturo Magidin for Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
Historically, when Leibniz conceived of the notation, $\frac{dy}{dx}$ was supposed to be a quotient: it was the quotient of the "infinitesimal change in $y$ produced by the change in $x$" divided by...
View ArticleAnswer by Anonymous for Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
The notation $dy/dx$ - in elementary calculus - is simply that: notation to denote the derivative of, in this case, $y$ w.r.t. $x$. (In this case $f'(x)$ is another notation to express essentially the...
View ArticleIs $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
In the book Thomas's Calculus (11th edition) it is mentioned (Section 3.8 pg 225) that the derivative $\frac{\textrm{d}y}{\textrm{d}x}$ is not a ratio. Couldn't it be interpreted as a ratio, because...
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