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I just read that if $v$ and $w$ are smooth functions of $x$ you can define operators $\partial _x + v$ and $\partial _x + w$. From there we find, $$ (\partial _x + v)(\partial _x + w) = \partial _x ^2 + w' + (v+w)\partial_x + vw \\ (\partial _x + v)(\partial _x + w) = \partial _x ^2 + v' + (v+w)\partial_x + vw $$ Where do these $w'$ and $v'$ terms come from? I do not follow the logic of how they appear.

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A really good way to make sure we don't miss terms like this is to use the method of a "test function." We have for some differentiable function $f(x)$ that

$$ (\partial_x+v)(\partial_x+w)f = (\partial_x+v)(\partial_xf + wf) = \partial^2_xf + v\partial_xf + \partial_x(wf) + vwf. $$

It's from the third term that your $w'$ comes out:

$$ (\partial_x+v)(\partial_x+w)f = (\partial_x+v)(\partial_xf + wf) = \partial^2_xf + v\partial_xf + w'f + w\partial_xf + vwf. $$

After evaluating fully, you can now say that since the final expression is true for "any" differentiable function $f$, the operator $(\partial_x+v)(\partial_x+w)$ is equivalent to the operator

$$ \partial^2_x + v\partial_x + w' + w\partial_x + vw. $$

The logic is the same for the other operator.

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