Source / pinned edition
The Chain Rule
Source of package openstax.calculus_derivatives · OpenStax Calculus, ch. 3 Derivatives, module m53581
- work
- urn:openstax:clir:calculus-derivatives#M53581
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- OpenStax Calculus, ch. 3 Derivatives, module m53581
- kind
- urn:openstax:clir:calculus-derivatives#treatise
m53581/index.cnxml at commit 8dbc2ce (main, 2026-07-15)
- edition
- urn:openstax:clir:calculus-derivatives#M53581_MAIN
- language
- en
- officiality
- official
- materialisation
- PINNED_OFFICIAL_BYTES
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- adopted: 2026-07-15
Publications
| URI | Type | documentHash | Retrieved |
|---|---|---|---|
| https://raw.githubusercontent.com/openstax/osbooks-calculus-bundle/8dbc2ce19e804924b2517b89ac72ee45be949d15/modules/m53581/index.cnxml | application/xml; charset=utf-8 | sha256:f848070e09084722cee9c21879d33b1873118bc7ec600d4c7f2bcd1e07693219 | 2026-09-12T00:00:00Z |
Fragments — 1
module/m53581#fs-id1169739194586 · section · referenced by: 1
en · official · sha256:cb428baef820847d90888ef3c2a814356c3c957cfa1a6758911973df198afb2d
<note id="fs-id1169739194586">
<title>Rule: The Chain Rule</title>
<para id="fs-id1169736612521">Let <m:math><m:mi>f</m:mi></m:math> and <m:math><m:mi>g</m:mi></m:math> be functions. For all <emphasis effect="italics">x</emphasis> in the domain of <m:math><m:mi>g</m:mi></m:math> for which <m:math><m:mi>g</m:mi></m:math> is differentiable at <emphasis effect="italics">x</emphasis> and <m:math><m:mi>f</m:mi></m:math> is differentiable at <m:math><m:mrow><m:mi>g</m:mi><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow><m:mo>,</m:mo></m:mrow></m:math> the derivative of the composite function</para>
<equation id="fs-id1169739233799" class="unnumbered"><label/><m:math><m:mrow><m:mi>h</m:mi><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow><m:mo>=</m:mo><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>f</m:mi><m:mo>∘</m:mo><m:mi>g</m:mi></m:mrow><m:mo>)</m:mo></m:mrow><m:mo stretchy="false">(</m:mo><m:mi>x</m:mi><m:mo stretchy="false">)</m:mo><m:mo>=</m:mo><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>g</m:mi><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow></m:mrow><m:mo>)</m:mo></m:mrow></m:mrow></m:math></equation>
<para id="fs-id1169738949353">is given by</para>
<equation id="fs-id1169738948938"><m:math><m:mrow><m:msup><m:mi>h</m:mi><m:mo>′</m:mo></m:msup><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow><m:mo>=</m:mo><m:msup><m:mi>f</m:mi><m:mo>′</m:mo></m:msup><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>g</m:mi><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow></m:mrow><m:mo>)</m:mo></m:mrow><m:msup><m:mi>g</m:mi><m:mo>′</m:mo></m:msup><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow><m:mo>.</m:mo></m:mrow></m:math></equation>
<para id="fs-id1169739199751">Alternatively, if <m:math><m:mi>y</m:mi></m:math> is a function of <m:math><m:mrow><m:mi>u</m:mi><m:mo>,</m:mo></m:mrow></m:math> and <m:math><m:mi>u</m:mi></m:math> is a function of <m:math><m:mrow><m:mi>x</m:mi><m:mo>,</m:mo></m:mrow></m:math> then</para>
<equation id="fs-id1169739187558" class="unnumbered"><label/><m:math><m:mrow><m:mfrac><m:mrow><m:mi>d</m:mi><m:mi>y</m:mi></m:mrow><m:mrow><m:mi>d</m:mi><m:mi>x</m:mi></m:mrow></m:mfrac><m:mo>=</m:mo><m:mfrac><m:mrow><m:mi>d</m:mi><m:mi>y</m:mi></m:mrow><m:mrow><m:mi>d</m:mi><m:mi>u</m:mi></m:mrow></m:mfrac><m:mo>·</m:mo><m:mfrac><m:mrow><m:mi>d</m:mi><m:mi>u</m:mi></m:mrow><m:mrow><m:mi>d</m:mi><m:mi>x</m:mi></m:mrow></m:mfrac><m:mo>.</m:mo></m:mrow></m:math></equation>
</note>Referencing norms: ChainRuleJustifies
The hash beside the text is the sha256 of its exact bytes. It is the check that the text shown matches the one pinned in the package: a divergence fails the oracle’s static analysis.