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The $\mathrm{d}x$ appears on integrals. I saw conflicting views regarding it. People sometimes write it does have a connection to differentiation and derivatives. Does it or does it not?

amWhy
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bilanush
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  • "Conflicting views" – bilanush Jul 30 '18 at 16:18
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    The simple answer to that concern is that they are not conflicting. You get a different answer depending on whether you ask an algebraist, a differential geometer, a physicist, or a probability theorist, sure. But they do not conflict, any more than one divided by two is either $\frac12$ or $0.5$, and that's not conflictive. The different answers convey different ways of thinking, but the resulting calculations will give the same answer. So yes, it does or does not have a connection to derivatives, and it's fine like that. – Arthur Jul 30 '18 at 16:22
  • in "TGIF", does the G stand for "Goodness" or "God"? Depends who's saying it, doesn't it? – John Hughes Jul 30 '18 at 16:22
  • Arthur What the heck ? I am talking about people writing its related to differentiation while other who deny it. – bilanush Jul 30 '18 at 16:45
  • That's probably because they're too stuck in their own interpretation to see that other interpretations are also true. – Arthur Jul 30 '18 at 16:48
  • It is used in expressing differential equations, @bilanush. Equations of the sort (a simple example): $\frac {dy}{dx} \cdot \frac 1{y} = (3x^2-2) \iff \frac {dy}{y} = (3x^2-2)dx$. This is solved by integrating the LHS with respect to y, and the RHS with respect to $x$. And so we obtain: $$\ln|y| = x^3-2x + c$$ or $$y= \exp(x^3-2x +C) = C\exp(x^3-2x)$$ – amWhy Jul 30 '18 at 17:30

2 Answers2

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$dx$ means $\Delta{x}$, or an infinitesimal change in $x$. The $\frac{d}{dx}$ in derivatives is simply a mathematical operator that you apply to a function; it comes from $\frac{\Delta{y}}{\Delta{x}}$ = $\frac{\Delta}{\Delta x}y=\frac{d}{dx}y$.

RayDansh
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  • Agree. But some people do appear to affiliate it with "differentiation" , "integration" what do they mean? – bilanush Jul 30 '18 at 16:19
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    For differentiation, we want the slope of the secant line to approximate the slope of the tangent line, so we want the change in $x$ and $y$ to become as small as possible. $\frac{dy}{dx}$ is used to represent this. For integrals, we are summing up rectangles under the curve (Riemann sum) as the width ($dx$) of these rectangles approaches $0$. So either way $dx$ means an infinitely small change in $x$. – RayDansh Jul 30 '18 at 16:23
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It all depends on the context in which $dx$ appears.

If $x$ is a function of $t$, then $$dx=x'(t)dt$$ involves derivative.

If $dx$ appears in an integral, it simply means integrate with respect to $x$.

If it appears as a denominator such as $\frac {df}{dx}$ it mean differentiate with respect to $x$

If it is standing by itself it means an increment in $x$

Mohammad Riazi-Kermani
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