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

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(Created page with "==Derivative of the Natural Exponential Function== Let <math>u</math> be a differentiable function of <math>x</math>. Then, 1.<math>\frac{d}{dx}[e^x]=e^x</math> 2.<math>...")
 
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   2.<math>\frac{d}{dx}[e^u]=e^u\frac{du}{dx}</math>
 
   2.<math>\frac{d}{dx}[e^u]=e^u\frac{du}{dx}</math>
  
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'''Exercises''' Differentiate each function:
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'''a)''' <math>f(x)=e^{2x}</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)=2e^{2x}</math>
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|}
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'''b)''' <math>f(x)=e^{3x^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)=6xe^{3x^2}</math>
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|}
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'''c)''' <math>f(x)=e^{-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)=-2xe^{2x}</math>
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|}
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'''d)''' <math>f(x)=4e^{-x}</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)=-4e^{-x}</math>
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|}
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'''e)''' <math>f(x)=\frac{e^x-e^{-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{e^x-e^{-x}}{2}=\frac{e^x}{2}-\frac{e^{-x}}{2}=\frac{1}{2}e^x-\frac{1}{2}e^{-x}</math>
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|-
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|<math>f'(x)=\frac{1}{2}e^x-\frac{1}{2}(-1)e^{-x}</math>
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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]]'''

Revision as of 07:39, 11 August 2020

Derivative of the Natural Exponential Function

 Let 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}
 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}
. Then,
 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}[e^x]=e^x}

 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}[e^u]=e^u\frac{du}{dx}}

Exercises Differentiate each 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)=e^{2x}}

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)=2e^{2x}}

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)=e^{3x^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)=6xe^{3x^2}}

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)=e^{-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)=-2xe^{2x}}

d) 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)=4e^{-x}}

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)=-4e^{-x}}

e) 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{e^x-e^{-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{e^x-e^{-x}}{2}=\frac{e^x}{2}-\frac{e^{-x}}{2}=\frac{1}{2}e^x-\frac{1}{2}e^{-x}}
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}{2}e^x-\frac{1}{2}(-1)e^{-x}}

Return to Topics Page

This page were made by Tri Phan

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