*Notes to a video lecture on http://www.unizor.com*

__Partial Derivatives -__

∂²/∂x ∂y = ∂²/∂y ∂x

∂²/∂x ∂y = ∂²/∂y ∂x

In the following examples compare

*∂²z/∂x ∂y*

*∂²z/∂y ∂x*

They should be identical.

*Example 1*

Let

**z=√x·y**Then

*∂*

**z**/∂**x**=**y/(2√x·y)***∂²*

**z**/∂**y**∂**x**=**1/(4√x·y)***∂*

**z**/∂**y**=**x/(2√x·y)***∂²*

**z**/∂**x**∂**y**=**1/(4√x·y)***Example 2*

Let

**z=e**^{x·y}Then

*∂*

**z**/∂**x**=**y·e**^{x·y}*∂²*

**z**/∂**y**∂**x**=**(x·y+1)·e**^{x·y}*∂*

**z**/∂**y**=**x·e**^{x·y}*∂²*

**z**/∂**x**∂**y**=**(x·y+1)·e**^{x·y}*Example 3*

Let

**z=1/(x²+y²)**Then

*∂*

**z**/∂**x**=**−2x/(x²+y²)²***∂²*

**z**/∂**y**∂**x**=**8x·y/(x²+y²)³***∂*

**z**/∂**y**=**−2y/(x²+y²)²***∂²*

**z**/∂**x**∂**y**=**8x·y/(x²+y²)³***Example 4*

Let

**z=sin(x)/y²**Then

*∂*

**z**/∂**x**=**cos(x)/y²***∂²*

**z**/∂**y**∂**x**=**−2cos(x)/y³***∂*

**z**/∂**y**=**−2sin(x)/y³***∂²*

**z**/∂**x**∂**y**=**−2cos(x)/y³***Example 5*

Let

**z=arctan(x√y)**Then

*∂*

**z**/∂**x**=**√y/(1+x²·y)***∂²*

**z**/∂**y**∂**x**=**(1−x²·y)/2√y(1+x²·y)²***∂*[

**z**/∂**y**=**x/***]*

**2√y(1+x²·y)***∂²*

**z**/∂**x**∂**y**=**(1−x²·y)/2√y(1+x²·y)²***Example 6*

Let

**z=y**^{x}Then

*∂*

**z**/∂**x**=**y**^{x}·ln(y)*∂²*

**z**/∂**y**∂**x**=**y**^{x−1}·(x·ln(y)+1)*∂*

**z**/∂**y**=**x·y**^{x−1}*∂²*

**z**/∂**x**∂**y**=**y**^{x−1}·(x·ln(y)+1)
## No comments:

Post a Comment