Math, asked by chanchal3858, 11 months ago

Explain Derivation of Integration with example

Answers

Answered by Anonymous

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⏭️Some mathematicians define a derivation as an operator such that

d(f+g)=df+dgd(f+g)=df+dg

d(λf)=λdfd(λf)=λdf

d(f⋅g)=df⋅g+f⋅dgd(f⋅g)=df⋅g+f⋅dg (Leibnitz rule)

And integration as the inverse operator, that is

∫f=g∫f=g, if and only if dg=fdg=f.

Well, basically if you can compute it, you've understood.

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Now, if you are not satisfied with a formal definition, you have a few intuitive interpretation.

For Newton, the derivation y' = dy/dx with respect to time is the speed.

For Liebnitz, the derivation is the variation of y when x varies of a very small amount.

For Bernouilli, the derivation is the tangent of a curve.

For Euler, the derivation is the first term of a serie expansion.

For Cartan, the derivation is the best linear approximation.

For Banach, the derivation is a limit.

For Gâteaux, the derivation measure the variation of density in a direction.

For Grothendieck, the derivation is computation that can be performed on some abstract spaces.

For Cavaleri, the integral is an area.

For Leibnitz and Newton, integration is a cumulative sum of very small quanties.

For Reimann, integration is a limit of sums.

For Lebesgue, integration is a measure.

For Weil, integration is a linear operator.

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Maybe a problem is that it is not easy to master all these definitions at start. Not to say to understand why they are the same.

I suggest you start with Newton or Leibnitz view, and be confident that, as long as you are using correctly the above computation rules, you will make no mistake in the other point of view, which will come slowly by slowly.

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Answered by sonuSiddiqui

Step-by-step explanation: