Definition of derivative - infinitesimal approach, help :)

[pondhockey]

Mark, thank you for the comment.

Can you tell me, in what context d (i.e. the differential) is considered an operator? It is not defined that way in either calculus/advance calculus text that I just now referred back to. In those texts, dx is a real variable, referred to as an independent variable, and dy is defined as f' dx. A comparable "total" differential is defined for functions of two variables. No "operator" or corresponding differential algebra is introduced - only standard algebra of real numbers is used.

 

[Mark44]

Mark, thank you for the comment.

Can you tell me, in what context d (i.e. the differential) is considered an operator? It is not defined that way in either calculus/advance calculus text that I just now referred back to. In those texts, dx is a real variable, referred to as an independent variable, and dy is defined as f' dx. A comparable "total" differential is defined for functions of two variables. No "operator" or corresponding differential algebra is introduced - only standard algebra of real numbers is used.

See https://en.wikipedia.org/wiki/Differential_of_a_function#Properties

 

[WWGD]

Yess I understand it! so if we choose any point fx (1,f(1)), and set dx or dy to any number fx if dx=5, then dy=10 (dy depends on dx) in our example is interpreted as: When the slope of the tangent f'(1)=2 at point (1,f(1)) is multiplied by the horizontal run of 5 units, then we "run along the tangent" and end up at the point (1+dx, f(1) + dy) = (1+5, 2+10) = (6,12) ON the tangent.

So would this be the correct Conclusion: So dy and dx can be calculated for bigger numbers/values of dy and dx, and then we end up on a more distant point on the tangent. However, in practice this is not useful, because we are interested in using dy to approximate the slope of the function f at a point near f'(x). But why bother using this method if we have the derivative of a function? Is it not more practical to get the 100% accurate slope af a point as dy/dx = f'(x)?

I don't know what you mean by 100% accurate, because this is a limit, involving $\delta$s and $\epsilon$s ; unless your graph is a straight line, this is not what I would call 100% accurate.

What you are trying to do is to approximate the _Real_ change in values of the function by the change along the tangent line. This is what dy=f'(x)dx gives you.

 

[WWGD]

Mark, thank you for the comment.

Can you tell me, in what context d (i.e. the differential) is considered an operator? It is not defined that way in either calculus/advance calculus text that I just now referred back to. In those texts, dx is a real variable, referred to as an independent variable, and dy is defined as f' dx. A comparable "total" differential is defined for functions of two variables. No "operator" or corresponding differential algebra is introduced - only standard algebra of real numbers is used.

But the idea is always the same: you have a tangent linear object (line, plane, etc.) that approximates the change of a (differentiable, at least at the point) function , at least near the point in question.

 

[pondhockey]

But the idea is always the same: you have a tangent linear object (line, plane, etc.) that approximates the change of a (differentiable, at least at the point) function , at least near the point in question.

I agree with this. And I think it begs the question of why mess with differentials in expositions of physics (and in particular thermodynamics.) To me it seems to cloud more issues than it simplifies. It certainly inspires a lot of threads just like this one!

 

[pondhockey]

Mark writes:

..

See https://en.wikipedia.org/wiki/Differential_of_a_function#Properties

..Thank you for the link. It makes me think that this info should be in an introduction to every physics and thermodynamics text. I've never seen it in an undergraduate math text. There's a culture, in physics, that I find regrettable: the prof used integrals and line (path) integrals in my physics class way before any of us were formally introduced to them. I think that handwaving and intuitive arguments become embedded in the culture and practice of physics.