Difference between revisions of "Math 22 Chain Rule"

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==The Chain Rule==
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  If <math>y=f(x)</math> is a differentiable function of <math>u</math> and <math>u=g(x)</math> is a
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  differentiable function of <math>x</math>, then <math>y=f(g(x))</math> is a differentiable function
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  of <math>x</math> and
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  <math>\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}</math>
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  In another word, <math>\frac{d}{dx}[f(g(x))]=f'(g(x))\cdot g'(x)</math>
  
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'''Example''': Find derivative of
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'''1)''' <math>f(x)=\sqrt{x^2+3x-4}</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 style="vertical-align: -5px">f(x)=f(x)=\sqrt{x^2+3x-4}=(x^2+3x-4)^{\frac{1}{2}}</math>
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|-
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|<math>f'(x)=\frac{1}{2}\cdot (x^2+3x-4)^{(\frac{1}{2} -1)}\frac{d}{dx}[x^2+3x-4]</math>
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|-
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|<math>=(x^2+3x-4)^{\frac{-1}{2}}(2x+3)</math>
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|}
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'''2)''' <math>f(x)=(x^2+1)^{100}</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 style="vertical-align: -5px">f'(x)=100(x^2+1)^{99} \frac{d}{dx}[x^2+1]</math>
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|-
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|<math>=100(x^2+1)^{99} (2x)</math>
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|}
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==The General Power Rule==
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  If <math>y=[u(x)]^n</math>, where <math>u</math> is a differentiable function of <math>x</math>
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  and <math>n</math> is a real number, then
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  <math>\frac{d}{dx}[u^n]=n\cdot u^{n-1}\cdot u'</math>
  
 
[[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 06:16, 23 July 2020

The Chain Rule

 If  is 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 u}
 and 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=g(x)}
 is 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 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 y=f(g(x))}
 is 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}
 and
 
 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{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}}

 
 In another word, 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}[f(g(x))]=f'(g(x))\cdot g'(x)}

Example: Find derivative of

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)=\sqrt{x^2+3x-4}}

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)=f(x)=\sqrt{x^2+3x-4}=(x^2+3x-4)^{\frac{1}{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)=\frac{1}{2}\cdot (x^2+3x-4)^{(\frac{1}{2} -1)}\frac{d}{dx}[x^2+3x-4]}
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^2+3x-4)^{\frac{-1}{2}}(2x+3)}

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)=(x^2+1)^{100}}

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)=100(x^2+1)^{99} \frac{d}{dx}[x^2+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 =100(x^2+1)^{99} (2x)}

The General Power Rule

 If 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 y=[u(x)]^n}
, where 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}
 is 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}
 
 and 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 n}
 is a real number, then
 
 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}[u^n]=n\cdot u^{n-1}\cdot u'}

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This page were made by Tri Phan

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