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Question about the differential in Calculus. Assume a function y = f(x) , differentiable everywhere. Now we have for some Δx

Δy = f(x + Δx) - f(x)

The differential of x, is defined as “dx”, can be any real number, and dx = Δx

The differential of y, is defined by “dy” and dy = f’(x) dx

Clearly,

Δy ≈ dy, depending on the magnitude of Δx.

In calculus an expression like “dx” usually denotes something infinitesimally small. Why is it necessary to have dy and dx used as real numbers of some magnitude? In specifying and solving calculus problems are not the usual symbols sufficient? Is it just a matter of notational convenience?

Agapito Martinez
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  • What are "the usual symbols"? – FShrike Jul 22 '21 at 21:23
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    $dx,dy$ are extremely usual – FShrike Jul 22 '21 at 21:23
  • Does this answer your question? – Arturo Magidin Jul 22 '21 at 21:26
  • Many thanks Monsieur Magidin for the link. Obviously the "dx" notation is useful practically in applying such results as the chain and inverse derivative rules. What is sometimes confusing is going from Δy ≈ f'(x) Δx (which makes sense) to dy = f'(x) dx which is not some deep result, but rather an expression defining both dy and dx. such that dx = Δx. Your historical review certainly explains the origin of this state of affairs, many thanks for making it available. – Agapito Martinez Jul 23 '21 at 18:03

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In my opinion, it is just to have an intuition that the derivative is the range of the tangent line. It also helps to understand "instantaneous velocity" when studying physics. The average velocity between time $t$ and time $t+\Delta t$ is given by

$$\frac{f(t+\Delta t)-f(t)}{\Delta t},$$

where $f$ is the position. So, it's quite intuitive that it gets closer and closer than the velocity at time $t$ as you decrease $\Delta t$.

Ygor Arthur
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