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christian0710

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Hi I'm reading

I do understand the basic idea: You make the secant between 2 points on a graph approach one of the points and at this point you get the tangent to the graph which is the derivative of f at that point and tells you the slope of the function at that point. But I want the correct mathematical understanding of it.

The book argues that

Then it defines that

and mentiones that

and prooves that

Is

This expression Δy/Δx only approaches f'(x): Δy/Δx = (y2-y1)/(x2-x1) = (f'(x)Δx+εΔx

While this expression is equal

The difference between

and

is that Δy= dy+ εΔx contains that extra "εΔx" term and therefore is bigger than dy and is also the secant between 2 points along the graph, whereas dy is an infinitesimal small movement between 2 points on the tangent to the graph. Because dy=f'(x)*Δx equals the term f'(x)*Δx this tells us that dy is an infinitesimally small change in y between the point at the tangent and another point on the tangent infinitesimally close to that point. The Δx in f'(x)Δx shows us that it's a "change in y" corresponding to a infinitesimally small change in x so (x2-x1), hence Δx?

If the

So my last question: I guess it's a no-go to treat the symbol dy/dx as a quotient? The book mentions that if dy=f'(x)dx and if dx ≠ 0 then we can rewrite the equation dy/dx = f'(x), is this just lucky that this "trick" works, or is it true that you can treat it like a quotient?

**Elementary calculus - an infinitesimal approach**and just wan't to make sure my understanding of what**dy, f'(x)**and**dx**means is correct.I do understand the basic idea: You make the secant between 2 points on a graph approach one of the points and at this point you get the tangent to the graph which is the derivative of f at that point and tells you the slope of the function at that point. But I want the correct mathematical understanding of it.

The book argues that

**Δy**= the change in y along a curve between 2 points (I assume it's a secant)**dy**= change in y along the tangent line to that curve between 2 points (The differential)Then it defines that

**Δy = f(x+Δx)-f(x)**

dy = f'(x)dy = f'(x)

**Δx**and mentiones that

*"let y=f(x). Suppose f'(x) exists at a cetain point x, and Δx is the infinitesimal, then Δy is infinitesimal and"***Δy = f'(x)Δx+ εΔx**and prooves that

**Δy/Δx ≈ f'(x)**

Δy/Δx =

My first question is This:

Δy/Δx =

**f'(x) + ε**

Δy =Δy =

**f'(x)Δx + εΔx**My first question is This:

Is

**f(x+Δx)-f(x)**and**Equal? If they both are equal to Δy then i assume they are?****f'(x)Δx+ εΔx****Is this the correct understanding:**This expression Δy/Δx only approaches f'(x): Δy/Δx = (y2-y1)/(x2-x1) = (f'(x)Δx+εΔx

**)**/((x+εΔx-x)) ≈ f'(x)While this expression is equal

**dy/dx = f'(x)**The difference between

**dy/dx = f'(x)**and

**Δy/Δx ≈ f'(x)**is that Δy= dy+ εΔx contains that extra "εΔx" term and therefore is bigger than dy and is also the secant between 2 points along the graph, whereas dy is an infinitesimal small movement between 2 points on the tangent to the graph. Because dy=f'(x)*Δx equals the term f'(x)*Δx this tells us that dy is an infinitesimally small change in y between the point at the tangent and another point on the tangent infinitesimally close to that point. The Δx in f'(x)Δx shows us that it's a "change in y" corresponding to a infinitesimally small change in x so (x2-x1), hence Δx?

If the

**f'(x)**exists then the differential**dy**and the increment**Δy**MUST be infitesimal and so close together that they cannot be seen under the infinitesimal microscope.So my last question: I guess it's a no-go to treat the symbol dy/dx as a quotient? The book mentions that if dy=f'(x)dx and if dx ≠ 0 then we can rewrite the equation dy/dx = f'(x), is this just lucky that this "trick" works, or is it true that you can treat it like a quotient?

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