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Someone said to me today:

You can solve any derivative using your standard derivative techniques like the product rule or chain rule, but you cannot solve every integral

Of course referring to derivatives that exist.

Is it true that I can do any derivative using my standard techniques? Why can't I do any integral with my standard techniques for integration? What's stopping me?

Featherball
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    The derivative of an elementary function is elementary. But for example the derivative of the gamma function can't be expressed in terms of the gamma function and elementary functions only. So I'd say not all derivatives can be "solved", if by solved one means "expressed in terms of other commonly known functions". – pregunton Dec 20 '19 at 11:30
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    Well, I think the "true" answer is the existence of the chain rule for differentiation and non-existance of the chain rule for integration. Of course there is no "elementary" way of differentiating $\Gamma$ function, but once you label this derivative somehow, you can differentiate any expression containing $\Gamma$ function – F. Jatpil Dec 20 '19 at 12:13
  • The answers in the other question don't answer what I was looking for. Why was my question closed? – Featherball Dec 20 '19 at 18:10

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