What constitutes a derivative?

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Suppose I have a small (infinitesimal) quantity $dy$ and another small quantity $dx$ and they are related by $dy = k \cdot dx$. Will that automatically imply that $\frac{dy}{dx}=k$ is the derivative $\left(\frac{\mathrm{d}y}{\mathrm{d}x}=\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x)- f(x)}{\Delta x}\right)$ of $x$ with respect to $y$?

I have seen several examples of such things occuring in engineering textbooks, such as electrical relations between the charge on a capacitor and the voltage across it. (I can't remember the details, and my notes are safely stored in the basement).

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Yes. You have y(x) = kx, and then $dy = \frac{\partial y}{\partial x} dx = \frac{\partial (kx)}{\partial x} dx = k dx$

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Suppose I have a small (infinitesimal) quantity $dy$ and another small quantity $dx$ and they are related by $dy = k \cdot dx$. Will that automatically imply that $\frac{dy}{dx}=k$ is the derivative $\left(\frac{\mathrm{d}y}{\mathrm{d}x}=\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x)- f(x)}{\Delta x}\right)$ of $x$ with respect to $y$?

I have seen several examples of such things occuring in engineering textbooks, such as electrical relations between the charge on a capacitor and the voltage across it. (I can't remember the details, and my notes are safely stored in the basement).

Given y = f(x) ,then we define : dy = f'(x)Δx and Δx= 1.dx .

Hence by definition: dy/dx = f'(x) ,where dy/dx is the ratio of the differentials dy ,dx

In the case where the derivative is denoted by :$\frac{dy}{dx}$,then this is equal with the ratio of the two differentials dy/dx

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