Why do all elementary functions have an elementary derivative? Considering many elementary functions have an antiderivative which is not elementary, why does this type of thing not also happen in differential calculus?

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*for the same reason that there are strict rules for grammar, but not for creating poems. Differentiation goes ‘downwards’ – in the direction of gravity, so to speak – whereas integration goes ‘upwards’ – against the pull of gravity, so to speak. The price of defeating gravity is ambiguity, which takes the form of a lack of strict rules. Notice that integration gives you a taste of ambiguity from the very start: changing the value of a function a single point does not change the value of the integral.

