# Difference between revisions of "Math 22 Partial Derivatives"

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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> | 4. <math>\frac{\partial}{\partial x}(\frac{\partial f}{\partial y})=\frac{\partial^2 f}{\partial x\partial y}=f_{yx}</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: | ||

+ | |- | ||

+ | |<math>f_x=4x-4y</math> | ||

+ | |- | ||

+ | |Then, <math>f_{xy}=-4</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: | ||

+ | |- | ||

+ | |<math>f_y=6xy-2+10x^2y</math> | ||

+ | |- | ||

+ | |Then, <math>f_{yx}=6y+20xy</math> | ||

+ | |} | ||

## Latest revision as of 17:21, 3 September 2020

## Partial Derivatives of a Function of Two Variables

If , 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: |
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**2)**

Solution: |
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**3)**

Solution: |
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(product rule +chain rule) |

## Higher-Order Partial Derivatives

1.

2.

3.

4.

**1)** Find , given that ,

Solution: |
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Then, |

**2)** Find , given that ,

Solution: |
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Then, |

**This page were made by Tri Phan**