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Derivatives of Exponential and Logarithmic Functions
Source of package openstax.calculus_derivatives · OpenStax Calculus, ch. 3 Derivatives, module m53586
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- urn:openstax:clir:calculus-derivatives#M53586
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- OpenStax Calculus, ch. 3 Derivatives, module m53586
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- urn:openstax:clir:calculus-derivatives#treatise
m53586/index.cnxml at commit 8dbc2ce (main, 2026-07-15)
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| URI | Type | documentHash | Retrieved |
|---|---|---|---|
| https://raw.githubusercontent.com/openstax/osbooks-calculus-bundle/8dbc2ce19e804924b2517b89ac72ee45be949d15/modules/m53586/index.cnxml | application/xml; charset=utf-8 | sha256:6ac66df1d00f39513463ac330c27d2cfc16fe1a504c00191c20bad32c92cfbd5 | 2026-09-12T00:00:00Z |
Fragments — 2
module/m53586#fs-id1169738226753 · section · referenced by: 1
en · official · sha256:cc4855776059fd9a1a3635e5356553fd15655d3f0cedf8e8fc8d0a9f56d8e3f5
<note id="fs-id1169738226753" class="theorem">
<title>Derivative of the Natural Exponential Function</title>
<para id="fs-id1169738220224">Let <m:math><m:mrow><m:mi>E</m:mi><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>e</m:mi><m:mi>x</m:mi></m:msup></m:mrow></m:math> be the natural exponential function. Then</para>
<equation id="fs-id1169737928243" class="unnumbered"><label/><m:math><m:mrow><m:msup><m:mi>E</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>e</m:mi><m:mi>x</m:mi></m:msup><m:mo>.</m:mo></m:mrow></m:math></equation>
<para id="fs-id1169737142141">In general,</para>
<equation id="fs-id1169738124964" class="unnumbered"><label/><m:math><m:mrow><m:mfrac><m:mi>d</m:mi><m:mrow><m:mi>d</m:mi><m:mi>x</m:mi></m:mrow></m:mfrac><m:mrow><m:mo>(</m:mo><m:mrow><m:msup><m:mi>e</m:mi><m:mrow><m:mi>g</m:mi><m:mo stretchy="false">(</m:mo><m:mi>x</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:msup></m:mrow><m:mo>)</m:mo></m:mrow><m:mo>=</m:mo><m:msup><m:mi>e</m:mi><m:mrow><m:mi>g</m:mi><m:mo stretchy="false">(</m:mo><m:mi>x</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:msup><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>
</note>Referencing norms: ExponentialJustifies
module/m53586#fs-id1169737927590 · section · referenced by: 1
en · official · sha256:845d0cb5cffc5397bd70da03574db5da0700c726ae5e8a7cbfdfe0b6cd42a77c
<note id="fs-id1169737927590" class="theorem">
<title>The Derivative of the Natural Logarithmic Function</title>
<para id="fs-id1169737911417">If <m:math><m:mrow><m:mi>x</m:mi><m:mo>></m:mo><m:mn>0</m:mn></m:mrow></m:math> and <m:math><m:mrow><m:mi>y</m:mi><m:mo>=</m:mo><m:mtext>ln</m:mtext><m:mspace width="0.1em"/><m:mi>x</m:mi><m:mo>,</m:mo></m:mrow></m:math> then</para>
<equation id="fs-id1169738223534"><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:mn>1</m:mn><m:mi>x</m:mi></m:mfrac><m:mo>.</m:mo></m:mrow></m:math></equation>
<para id="fs-id1169737765282">More generally, let <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:mrow></m:math> be a differentiable function. For all values of <m:math><m:mi>x</m:mi></m:math> for which <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:mn>0</m:mn><m:mo>,</m:mo></m:mrow></m:math> the derivative of <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:mtext>ln</m:mtext><m:mspace width="0.1em"/><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> is given by</para>
<equation id="fs-id1169737919348"><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:mfrac><m:mn>1</m:mn><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:mfrac><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>
</note>Referencing norms: LogarithmJustifies
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