Difference between revisions of "Math 22 Derivatives of Logarithmic Functions"

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   2.<math>\frac{d}{dx}[\ln u]=\frac{1}{u}\frac{du}{dx}</math> for <math>u>0</math>
 
   2.<math>\frac{d}{dx}[\ln u]=\frac{1}{u}\frac{du}{dx}</math> for <math>u>0</math>
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'''Exercises''' Find the derivative of the function
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'''a)''' <math>f(x)=\ln(7x)</math>
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{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
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!Solution: &nbsp;
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|-
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|<math>f'(x)=\frac{1}{7x} (7x)'=\frac{1}{7x}7=\frac{1}{x}</math>
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|}
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'''b)''' <math>f(x)=\ln (x^8)</math>
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{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
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!Solution: &nbsp;
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|-
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|Solution 1: <math>f'(x)=\frac{1}{x^8}(x^8)'=\frac{8x^7}{x^8}=\frac{8}{x}</math>
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|-
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|Solution 2: <math>f(x)=\ln (x^8)=8\ln x</math>, so <math>f'(x)=8\frac{1}{x}=\frac{8}{x}</math>
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|}
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'''c)''' <math>f(x)=\ln (4-x^2)</math>
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{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
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!Solution: &nbsp;
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|-
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|<math>f'(x)=\frac{1}{4-x^2}(4-x^2)'=\frac{1}{4-x^2}(-2x)=\frac{-2x}{4-x^2}</math>
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|}
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[[Math_22| '''Return to Topics Page''']]
 
[[Math_22| '''Return to Topics Page''']]
  
 
'''This page were made by [[Contributors|Tri Phan]]'''
 
'''This page were made by [[Contributors|Tri Phan]]'''

Latest revision as of 09:07, 11 August 2020

Derivative of the Natural Logarithmic Function

 Let  be a differentiable function of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x}
.
 1.Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx}[\ln x]=\frac{1}{x}}
 for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x>0}

 
 2.Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx}[\ln u]=\frac{1}{u}\frac{du}{dx}}
 for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u>0}

Exercises Find the derivative of the function

a) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=\ln(7x)}

Solution:  
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x)=\frac{1}{7x} (7x)'=\frac{1}{7x}7=\frac{1}{x}}

b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=\ln (x^8)}

Solution:  
Solution 1: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x)=\frac{1}{x^8}(x^8)'=\frac{8x^7}{x^8}=\frac{8}{x}}
Solution 2: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=\ln (x^8)=8\ln x} , so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x)=8\frac{1}{x}=\frac{8}{x}}

c) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=\ln (4-x^2)}

Solution:  
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x)=\frac{1}{4-x^2}(4-x^2)'=\frac{1}{4-x^2}(-2x)=\frac{-2x}{4-x^2}}


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This page were made by Tri Phan

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