Source / pinned edition
Defining the Derivative
Source of package openstax.calculus_derivatives · OpenStax Calculus, ch. 3 Derivatives, module m53495
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- urn:openstax:clir:calculus-derivatives#M53495
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- OpenStax Calculus, ch. 3 Derivatives, module m53495
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- urn:openstax:clir:calculus-derivatives#treatise
m53495/index.cnxml at commit 8dbc2ce (main, 2026-07-15)
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- urn:openstax:clir:calculus-derivatives#M53495_MAIN
- language
- en
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- official
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- adopted: 2026-07-15
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| https://raw.githubusercontent.com/openstax/osbooks-calculus-bundle/8dbc2ce19e804924b2517b89ac72ee45be949d15/modules/m53495/index.cnxml | application/xml; charset=utf-8 | sha256:d0668acbea9b24dfe647e4383e48b1b21c630f21b503883ef1e94eaa7406a4bf | 2026-09-12T00:00:00Z |
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module/m53495#fs-id1169739095772 · defn · referenced by: 7
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<note id="fs-id1169739095772">
<title>Definition</title>
<para id="fs-id1169738924962">Let <m:math><m:mi>f</m:mi></m:math> be a function defined on an interval <m:math><m:mi>I</m:mi></m:math> containing <m:math><m:mi>a</m:mi><m:mo>.</m:mo></m:math> If <m:math><m:mrow><m:mi>x</m:mi><m:mo>≠</m:mo><m:mi>a</m:mi></m:mrow></m:math> is in <m:math><m:mi>I</m:mi><m:mo>,</m:mo></m:math> then</para>
<equation id="fs-id1169738850293"><m:math><m:mrow><m:mi>Q</m:mi><m:mo>=</m:mo><m:mfrac><m:mrow><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow><m:mo>−</m:mo><m:mi>f</m:mi><m:mo stretchy="false">(</m:mo><m:mi>a</m:mi><m:mo stretchy="false">)</m:mo></m:mrow><m:mrow><m:mi>x</m:mi><m:mo>−</m:mo><m:mi>a</m:mi></m:mrow></m:mfrac></m:mrow></m:math></equation>
<para id="fs-id1169738879993">is a <term id="term-00003">difference quotient</term>.</para>
<para id="fs-id1169739195558">Also, if <m:math><m:mrow><m:mi>h</m:mi><m:mo>≠</m:mo><m:mn>0</m:mn></m:mrow></m:math> is chosen so that <m:math><m:mrow><m:mi>a</m:mi><m:mo>+</m:mo><m:mi>h</m:mi></m:mrow></m:math> is in <m:math><m:mrow><m:mi>I</m:mi><m:mo>,</m:mo></m:mrow></m:math> then</para>
<equation id="fs-id1169739097936"><m:math><m:mrow><m:mi>Q</m:mi><m:mo>=</m:mo><m:mfrac><m:mrow><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>a</m:mi><m:mo>+</m:mo><m:mi>h</m:mi></m:mrow><m:mo>)</m:mo></m:mrow><m:mo>−</m:mo><m:mi>f</m:mi><m:mo stretchy="false">(</m:mo><m:mi>a</m:mi><m:mo stretchy="false">)</m:mo></m:mrow><m:mi>h</m:mi></m:mfrac></m:mrow></m:math></equation>
<para id="fs-id1169738913121">is a difference quotient with increment <m:math><m:mi>h</m:mi><m:mo>.</m:mo></m:math></para>
</note>Referencing norms: DifferentiationRule, InstructionAvailable, instruction_available, rule_conclusion, rule_method, rule_premise, step_justified_by
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.