What is the Precise Meaning of dA=2xdx?

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The discussion centers on the mathematical expressions dA=2xdx and dA/dx=2x, exploring their meanings and the validity of transitioning between them. It clarifies that dA/dx represents the derivative, indicating the instantaneous rate of change of area A with respect to side length x. The expression dA=2xdx is valid as it represents the differential form, linking changes in area to changes in side length. Additionally, the conversation touches on the definition of differentials and their interpretations in calculus.

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  • Understanding of calculus concepts, specifically derivatives and differentials.
  • Familiarity with the notation and interpretation of dA/dx in calculus.
  • Knowledge of basic geometric principles, particularly the area of a square.
  • Ability to interpret mathematical expressions and their transformations.
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  • Research the formal definitions of derivatives and differentials in calculus.
  • Study the relationship between differentials and integrals in calculus.
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lamsung
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Every equation has a meaning.

For example, the area of a square with side x is given by A=x2.

Then dA/dx can be computed and is equal to 2x.

The expression dA/dx=2x means the instantaneous change of A when x increases from a is 2a.

However, what is the meaning of dA=2xdx?

dA/dx is not a fraction, why is it valid to say dA=2xdx from the premise dA/dx=2x?

Furthermore, can anyone explain the same questions for dx=dA/2x?
 
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lamsung said:
Every equation has a meaning.

For example, the area of a square with side x is given by A=x2.

Then dA/dx can be computed and is equal to 2x.

The expression dA/dx=2x means the instantaneous change of A when x increases from a is 2a.

However, what is the meaning of dA=2xdx?

dA/dx is not a fraction, why is it valid to say dA=2xdx from the premise dA/dx=2x?

Furthermore, can anyone explain the same questions for dx=dA/2x?

Hi lamsung, :)

Welcome to MHB! (Handshake)

The derivative and the differential can be separately defined and there are several approaches to define the differential precisely. You would find those definitions >>here<<. However a more simple (and less precise) definition for the differential can be found >>here<<.
 

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