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<p>which is in general impossible to define for a <a href="page.php?w=ring_%28mathematics%29">ring</a>.  Many of the properties of the derivative are true of the formal derivative, but some, especially those that make numerical statements, are not.</p>

<p>Formal differentiation is used in algebra to test for <a href="page.php?w=multiple_roots_of_a_polynomial">multiple roots of a polynomial</a>.</p>

<p><big>Definition</big></p>
<p>Fix a ring  (not necessarily commutative) and let  be the ring of polynomials over .(If  is not commutative, this is the <a href="page.php?w=free_algebra">free algebra</a></p><p>
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