If is differentiable at the point and is differentiable at the point , then is differentiable at . Furthermore, let and , then
(1)
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There are a number of related results that also go under the name of "chain rules." For example, if , , and , then
(2)
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The "general" chain rule applies to two sets of functions
(3)
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(4)
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(5)
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and
(6)
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(7)
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(8)
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Defining the Jacobi rotation matrix by
(9)
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and similarly for and , then gives
(10)
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In differential form, this becomes
(11)
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(Kaplan 1984).