Articles of differential

What does it mean when dx is put on the start in an integral?

This question already has an answer here: Notation: Why write the differential first? 2 answers

derivative $\dfrac{d\left\{B(A_1, \ldots, A_M)\right\}}{d\left\{C(A_1, \ldots, A_M)\right\}}$ equals what?

Question: \begin{equation} \dfrac{dB}{dC} = ? \end{equation} Context The present problem might be considered a practical exercise with respect to the ongoing discussion in [1]. In addition, [2] offers a related question in that both the question here and there are derivatives with respect to a function. Given: \begin{align} A & = f{\left(A_1, \ldots, A_M\right)} = […]

Continuity of Derivative at a point.

Is it possible that derivative of a function exists at a point but derivative does not exist in neighbourhood of that point. If this happens then how is it possible. I feel that if derivative exists at a point then the left hand derivative is equal to the right hand derivative so derivative should exist […]

What is the logic behind decomposing a derivative operator symbol. In the population growth equation?

This question already has an answer here: What am I doing when I separate the variables of a differential equation? 4 answers

Sum of square of function

If $f'(x) = g(x)$ and $g'(x) = – f(x)$ for all real $x$ and $f(5) =2 =f'(5)$ then we have to find $f^2$$(10) + g^2(10)$ I tried but got stuck

Notation regarding different derivatives

I am currently reading up on partial derivatives and differentials in general. And there are a few points that seem unlcear to me (notation-wise). For example, if $f:\mathbb R\to\mathbb R,x\mapsto f(x)$ is a function, then is the following notation correct? $$\frac{d}{dx}f(x)=\frac{df}{dx}(x)=\frac{\partial}{\partial x}f(x)=\frac{\partial f}{\partial x}(x)=f'(x)$$ Now, in one of my lectures we wrote $$g(x):=\frac{\partial \log (f(x))}{\partial […]

When can't $dy/dx$ be used as a ratio/fraction?

By searching this question, I found this: Can I ever go wrong if I keep thinking of derivatives as ratios? However, the answers don’t have what I’m looking for! (Edit: Meaning, a counterexample. There is one involving partial derivatives, but then the only difference has to do with signs, which means that $dy/dx$ can still […]

Differentials Definition

Please define differentials rigorously such that they give a consistency to their use in the following links. I have read Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio? What is the practical difference between a differential and a derivative? Differential of a function at Wikipedia. If $\frac{dy}{dt}dt$ doesn't cancel, then what do you call it? Leibniz’s notation at […]

Rigorous definition of “Differential”

When it comes to definitions, I will be very strict. Most textbooks tend to define differential of a function/variable like this: Let $f(x)$ be a differentiable function. By assuming that changes in $x$ are small enough, we can say: $$\Delta f(x)\approx {f}'(x)\Delta x$$ Where $\Delta f(x)$ is the changes in the value of function. Now […]

Why it is absolutely mistaken to cancel out differentials?

In many of my physics courses, (don’t worry, this is a mathematics question!) My teachers cancel out differentials, and every time, they say: “If a mathematician saw me canceling out this differentials he would get mad, but we are physicists so, there is no problem in doing this.” However, still, I haven’t seen an acceptable […]