# derivative (S f g) ?

Conal Elliott conal@microsoft.com
Tue, 21 May 2002 10:54:32 -0700

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This isn't really a Haskell question, but I'm hoping a fellow Haskeller

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Has anyone seen a generalization of the chain rule for derivatives that
applies to applications of the S combinator?  The conventional chain
rule applies to the more restricted composition combinator:

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D (f . g) =3D (D f . g) * D g

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Where D is the differentiation higher-order function, "*" is multiply
lifted pointwise to functions (\ a b x -> a x * b x), and "." is
function composition.

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Thanks,

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- Conal

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<p class=3DMsoNormal><font size=3D2 face=3DArial><span =
style=3D'font-size:10.0pt;
font-family:Arial'>This isn&#8217;t really a Haskell question, but =
I&#8217;m
pointers.</span></font></p>

<p class=3DMsoNormal><font size=3D2 face=3DArial><span =
style=3D'font-size:10.0pt;
font-family:Arial'>&nbsp;</span></font></p>

<p class=3DMsoNormal><font size=3D2 face=3DArial><span =
style=3D'font-size:10.0pt;
font-family:Arial'>Has anyone seen a generalization of the chain rule =
for
derivatives that applies to applications of the S combinator?&nbsp; The
conventional chain rule applies to the more restricted composition =
combinator:</span></font></p>

<p class=3DMsoNormal><font size=3D2 face=3DArial><span =
style=3D'font-size:10.0pt;
font-family:Arial'>&nbsp;</span></font></p>

<p class=3DMsoNormal><font size=3D2 face=3DArial><span =
style=3D'font-size:10.0pt;
font-family:Arial'>&nbsp; D (f . g) =3D (D f . g) * D =
g</span></font></p>

<p class=3DMsoNormal><font size=3D2 face=3DArial><span =
style=3D'font-size:10.0pt;
font-family:Arial'>&nbsp;</span></font></p>

<p class=3DMsoNormal><font size=3D2 face=3DArial><span =
style=3D'font-size:10.0pt;
font-family:Arial'>Where D is the differentiation higher-order function, =
&#8220;*&#8221;
is multiply lifted pointwise to functions (\ a b x -&gt; a x * b x), and =
&#8220;.&#8221;
is function composition.</span></font></p>

<p class=3DMsoNormal><font size=3D2 face=3DArial><span =
style=3D'font-size:10.0pt;
font-family:Arial'>&nbsp;</span></font></p>

<p class=3DMsoNormal><font size=3D2 face=3DArial><span =
style=3D'font-size:10.0pt;
font-family:Arial'>Thanks,</span></font></p>

<p class=3DMsoNormal><font size=3D2 face=3DArial><span =
style=3D'font-size:10.0pt;
font-family:Arial'>&nbsp;</span></font></p>

<p class=3DMsoNormal><font size=3D3 face=3D"Times New Roman"><span =
style=3D'font-size:
12.0pt'>&nbsp;&nbsp;&nbsp; - Conal</span></font></p>

<p class=3DMsoNormal><font size=3D3 face=3D"Times New Roman"><span =
style=3D'font-size:
12.0pt'>&nbsp;</span></font></p>

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