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Here's something I've been trying to wrap my head around:

In general, it seems that integration is 'harder' than differentiation. At least analytically. Numerically it may be the other way round.

For one thing, it's often easy to differentiate implicit functions. For example, exp(y)+y=x implicitly defines a function y(x), even though we can't solve it explicitly with elementary functions. But we can still differentiate it implicitly by differentiating both sides for x:

dy/dx exp(y)+dy/dx=1, and solving for

dy/dx=1/(exp(y)+1)

So this is an analytical solution for dy/dx, although it depends on knowledge of y to compute its value.

As a side note, would we call this an explicit solution for dy/dx? I'm not sure, since y appears on the right side (but dy/dx does not).

But what I've really been wondering about is that as far as I know, there is no way to implicitly integrate y from this equation. It seems that we would need prior knowledge of y to do this.

Is this correct, and there is no general way to integrate an implicit function?

But what is really the fundamental reason for this differences between differentiating and integrating? On some level it seems to me that in differentiating we are discarding some information, so in a sense it's easier and requires less knowledge of the mathematical relationships.

In integration, I'm not sure we can quite say the opposite ('adding' information, except in the sense of the integration constant), but we're not discarding any.

So perhaps that's one way of looking at the difference, and the reason why integration is 'harder'.

But it's not really a very satisfying and concrete answer.

I know this is a bit of a vague post, but does anyone have any thoughts on the topic?

Sincerely,

Not Euler

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# Digging deeper into (implicit) differentiation and integration

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