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Every text that I read starts by defining differentiation then integration... but does anyone know if there is one that goes the other way? Also is there any harm in taking this approach.... to me, the current approach is only done because it has always been this way. I say this because I believe that this approach would allow a neater flow of logic in teaching calculus.

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I think the idea behind this is that differentiation is usually far easier than integration and allows the practical introduction of limits easier. –  George V. Williams Jan 19 '13 at 21:12
    
How would you integrate by parts without differentiation? Too much trouble, I think. –  Git Gud Jan 19 '13 at 21:13
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Apostol's Calculus text books go through integration before differentiation. –  Mike B Jan 19 '13 at 21:37

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Richard Courant's Differential and Integral Calculus discusses the integral before differentiation.

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as Mike noted: Apostol, Calculus (1967) begins with integration.

But then, later, derivatives are defined in the usual way, not somehow obtained as anti-integral.

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