2015年5月2日 · The symbols d/dx and dy/dx represent derivatives in calculus, indicating rates of change with respect to a variable.

We write $\frac{dy}{dx}$ but this is just notational, convention really. First, it is important to remember that this is not a ratio (see this, which is an excellent discussion of $\frac{dy}{dx}$), it is a limit and there is a limit definition, see the brief section here for an idea. The idea is we approximate the change of functions using an ...

2024年9月16日 · So we have (dy/dx)(x^2)=(2xi+i^2)/i. The denominator "i" comes as an infinitesimal by definition. The square of an infinitesimal, and the product of a real (or finite number) and an infinitesimal comes as an infinitesimal (see Keisler's book in anon's statement).

Steps for Using {eq}dy/dx {/eq} as a Notation for the Derivative of {eq}y = f(x) {/eq} Step 1: Determine what derivative rule, if any, is necessary to take the derivative of the function f(x ...

In differential calculus, We know that dy/dx is the ratio between the change in y and the change in x. In other words, the rate of change in y with respect to x.

2017年2月9日 · $\frac{dy}{dx}$ is, by definition, the limit of a secant line as the distance between two points approaches zero - it simply is the slope, nothing more to prove really (other than that the derivative actually exists, which is beyond the scope of this question)

2023年8月3日 · I understand in the case of $\frac{dy}{dx}$ or $\int f(x)dx$, but on their own, what is the formal definition? What is the intuition behind an infinitesimally small change in x (or y) being multiplied by a function?

2018年8月8日 · Well the derivative is given by: $$\lim_{dx \to 0} \frac{f(x+dx)-f(x)}{dx}=\lim_{dx\to 0} \frac{dy}{dx}$$ By definition the derivative is the rate of change of y with regard to x. That's why RHS stands.

2011年3月5日 · In Leibniz notation, the 2nd derivative is written as $$\dfrac{\mathrm d^2y}{\mathrm dx^2}\ ?$$ Why is the location of the $2$ in different places in the $\mathrm dy/\mathrm dx$ terms?

2018年7月29日 · Let me start with a preface that, to really get into the "true" rigorous definitions of $\text dx$ and $\text dy$, one needs to have multivariate calculus and linear algebra as a prerequisite, and should study "differential geometry", which is the mathematical framework that uses these objects in a rigorous manner.