Can I always think of derivatives this way?

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In summary, the conversation discusses the concept of differentials and how they are treated in physics. One person explains that differentials are seen as small changes or elements in a function, while another person questions if this is always a valid way to think about derivatives. The problem with this infinitesimal picture is also discussed, with one person pointing out the issue with having two quantities on the left but only one limit on the right. Two links to recent discussions on this topic are also provided.
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Ahmed1029
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Can I always treat derivatives as a small element lf y divided by a small element of x?
In physics, the differential is always treated as a little change or a tiny element of something, be it volume, area, etc. However, when I differentiate a function with respect to another, I always of it as a change divided by a change, not an element divided by an element: like when volume is an explicit function of area, I think of how much the volume changes, rather than taking a small element of volume and dividing it by a small element of area. My question is : are the two "definitions" equally valid? Can I always think of a derivative both ways? + I only know calculus and elementary differential equations so I will probably not understand any answer given in terms of differential forms, so please keep it simple 🥲
 
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The problem with the infinitesimal picture is that ##\dfrac{dx}{dt}=\displaystyle{\lim_{t\to 0}\dfrac{x(t_0+t)-x(t_0)}{t}}## has two quantities on the left but only one limit on the right. Pretending ##dx,dt## were two limits gets immediately wrong.
 
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1. What is the concept of derivatives?

The concept of derivatives is a fundamental idea in calculus that represents the rate of change of a function at a specific point. It measures how much a function changes with respect to its input variable.

2. Can derivatives always be thought of as slopes?

Yes, derivatives can always be thought of as slopes because they represent the rate of change of a function at a specific point, which is equivalent to the slope of the tangent line to the function at that point.

3. Are there other ways to think about derivatives?

Yes, there are other ways to think about derivatives, such as the instantaneous rate of change, the velocity of an object, and the acceleration of an object. These interpretations are all related to the concept of derivatives.

4. How are derivatives used in real-world applications?

Derivatives are used in various real-world applications, such as physics, economics, engineering, and finance. They are used to model and analyze the rates of change of quantities, such as velocity, acceleration, and interest rates.

5. Is it important to understand derivatives for non-mathematical fields?

Yes, understanding derivatives is important for non-mathematical fields because they are used in various real-world applications and can provide valuable insights and solutions to problems. They also help in understanding the behavior and relationships between variables in different fields of study.

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