Math 22 Partial Derivatives - Revision history

Tphan046: /* Note */

2020-09-04T00:21:27Z

Note

2020-08-18T17:15:12Z

2020-08-18T15:48:00Z

Higher-Order Partial Derivatives

2020-08-18T15:47:20Z

2020-08-18T15:42:31Z

2020-08-18T15:40:49Z

Higher-Order Partial Derivatives

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2020-08-18T15:38:19Z

2020-08-18T15:37:57Z

2020-08-18T15:36:12Z

@@ Line 62: / Line 62: @@
 |Then, <math>f_{yx}=6y+20xy</math>
 |}
-==Note==

@@ Line 62: / Line 62: @@
 |Then, <math>f_{yx}=6y+20xy</math>
 |}
+==Note==

@@ Line 45: / Line 45: @@
 . <math>\frac{\partial}{\partial x}(\frac{\partial f}{\partial y})=\frac{\partial^2 f}{\partial x\partial y}=f_{yx}</math>
-'''1)''' <math>f(x,y)=2x^2-4xy</math>, find <math>f_{xy}</math>
+'''1)''' Find <math>f_{xy}</math>, given that <math>f(x,y)=2x^2-4xy</math>,
 {| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
 !Solution: &nbsp;
@@ Line 54: / Line 54: @@
 |}
-'''2)''' <math>z=f(x,y)=3xy^2-2y+5x^2y^2</math>, find <math>f_{yx}</math>
+'''2)''' Find <math>f_{yx}</math>, given that <math>z=f(x,y)=3xy^2-2y+5x^2y^2</math>,
 {| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
 !Solution: &nbsp;

← Older revision		Revision as of 15:47, 18 August 2020
Line 45:		Line 45:
	4. <math>\frac{\partial}{\partial x}(\frac{\partial f}{\partial y})=\frac{\partial^2 f}{\partial x\partial y}=f_{yx}</math>		4. <math>\frac{\partial}{\partial x}(\frac{\partial f}{\partial y})=\frac{\partial^2 f}{\partial x\partial y}=f_{yx}</math>

		+	'''1)''' <math>f(x,y)=2x^2-4xy</math>, find <math>f_{xy}</math>
		+	{\| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
		+	!Solution:
		+	\|-
		+	\|<math>f_x=4x-4y</math>
		+	\|-
		+	\|Then, <math>f_{xy}=-4</math>
		+	\|}

−		+	'''2)''' <math>z=f(x,y)=3xy^2-2y+5x^2y^2</math>, find <math>f_{yx}</math>
		+	{\| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
		+	!Solution:
		+	\|-
		+	\|<math>f_y=6xy-2+10x^2y</math>
		+	\|-
		+	\|Then, <math>f_{yx}=6y+20xy</math>
		+	\|}

@@ Line 38: / Line 38: @@
 ==Higher-Order Partial Derivatives==
 . <math>\frac{\partial}{\partial x}(\frac{\partial f}{\partial x})=\frac{\partial^2 f}{\partial x^2}=f_{xx}</math>
 [[Math_22| '''Return to Topics Page''']]
 '''This page were made by [[Contributors|Tri Phan]]'''

@@ Line 33: / Line 33: @@
 |<math>\frac{\partial z}{\partial x}=2xe^{x^2y}+x^2e^{x^2y}2xy</math> (product rule +chain rule)
 |-
-|<math>\frac{\partial z}{\partial y}=x^2e^{x^2y}x^2</math>
+|<math>\frac{\partial z}{\partial y}=x^2e^{x^2y}(x^2)=x^4e^{x^2y}</math>
 |}

@@ Line 18: / Line 18: @@
 |}
-'''1)''' <math>z=f(x,y)=x^2y^3</math>
+'''2)''' <math>z=f(x,y)=x^2y^3</math>
 {| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
 !Solution: &nbsp;
@@ Line 27: / Line 27: @@
 |}
+'''3)''' <math>z=f(x,y)=x^2e^{x^2y}</math>

@@ Line 22: / Line 22: @@
 !Solution: &nbsp;
 |-
-|<math>\frac{\partial z}{\partial x}=4x^2-4y</math>
+|<math>\frac{\partial z}{\partial x}=2xy^3</math>
 |-
-|<math>\frac{\partial z}{\partial y}=-4x</math>
+|<math>\frac{\partial z}{\partial y}=3x^2y^2</math>
 |}