Difference between revisions of "Math 22 Partial Derivatives"

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   If <math>z=f(x,y)</math>, then the first partial derivatives of  with respect to <math>x</math> and <math>y</math> are the functions <math>\frac{\partial z}{\partial x}</math> and <math>\frac{\partial z}{\partial x}</math>, defined as shown.
 
   If <math>z=f(x,y)</math>, then the first partial derivatives of  with respect to <math>x</math> and <math>y</math> are the functions <math>\frac{\partial z}{\partial x}</math> and <math>\frac{\partial z}{\partial x}</math>, defined as shown.
 
    
 
    
   <math>\frac{\partial z}{\partial x}=\lim_{\delta x\to 0}\frac{f(x+\delta x,y)-f(x,y)}{\delta x}</math>
+
   <math>\frac{\partial z}{\partial x}=\lim_{\Delta x\to 0}\frac{f(x+\Delta x,y)-f(x,y)}{\Delta x}</math>
 
    
 
    
   <math>\frac{\partial z}{\partial y}=\lim_{\delta y\to 0}\frac{f(x,y+\delta y)-f(x,y)}{\delta y}</math>
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   <math>\frac{\partial z}{\partial y}=\lim_{\Delta y\to 0}\frac{f(x,y+\Delta y)-f(x,y)}{\Delta y}</math>
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 +
  We can denote <math>\frac{\partial z}{\partial x}</math> as <math>f_x(x,y)</math> and <math>\frac{\partial z}{\partial y}</math> as <math>f_y(x,y)</math>
 +
'''Example:''' Find <math>\frac{\partial z}{\partial x}</math> and <math>\frac{\partial z}{\partial y}</math> of:
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'''1)''' <math>z=f(x,y)=2x^2-4xy</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>\frac{\partial z}{\partial x}=4x^2-4y</math>
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|-
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|<math>\frac{\partial z}{\partial y}=-4x</math>
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|}
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'''2)''' <math>z=f(x,y)=x^2y^3</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>\frac{\partial z}{\partial x}=2xy^3</math>
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|-
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|<math>\frac{\partial z}{\partial y}=3x^2y^2</math>
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|}
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'''3)''' <math>z=f(x,y)=x^2e^{x^2y}</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>\frac{\partial z}{\partial x}=2xe^{x^2y}+x^2e^{x^2y}2xy</math> (product rule +chain rule)
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|-
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|<math>\frac{\partial z}{\partial y}=x^2e^{x^2y}(x^2)=x^4e^{x^2y}</math>
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|}
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 +
==Higher-Order Partial Derivatives==
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1. <math>\frac{\partial}{\partial x}(\frac{\partial f}{\partial x})=\frac{\partial^2 f}{\partial x^2}=f_{xx}</math>
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2. <math>\frac{\partial}{\partial y}(\frac{\partial f}{\partial y})=\frac{\partial^2 f}{\partial y^2}=f_{yy}</math>
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3. <math>\frac{\partial}{\partial y}(\frac{\partial f}{\partial x})=\frac{\partial^2 f}{\partial y\partial x}=f_{xy}</math>
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4. <math>\frac{\partial}{\partial x}(\frac{\partial f}{\partial y})=\frac{\partial^2 f}{\partial x\partial y}=f_{yx}</math>
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 +
'''1)''' Find <math>f_{xy}</math>, given that <math>f(x,y)=2x^2-4xy</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=4x-4y</math>
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|-
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|Then, <math>f_{xy}=-4</math>
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|}
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'''2)''' Find <math>f_{yx}</math>, given that <math>z=f(x,y)=3xy^2-2y+5x^2y^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_y=6xy-2+10x^2y</math>
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|-
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|Then, <math>f_{yx}=6y+20xy</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 16:21, 3 September 2020

Partial Derivatives of a Function of Two Variables

 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 z=f(x,y)}
, then the first partial derivatives of  with respect to  and  are the functions  and , defined as shown.
 
 
 
 
 
 We can denote  as  and  as 

Example: Find and of:

1)

Solution:  

2)

Solution:  

3)

Solution:  
(product rule +chain rule)

Higher-Order Partial Derivatives

1.

2.

3.

4.

1) Find , given that ,

Solution:  
Then,

2) Find , given that ,

Solution:  
Then,



Return to Topics Page

This page were made by Tri Phan