Difference between revisions of "Math 22 Partial Derivatives"

Partial Derivatives of a Function of Two Variables

If $z=f(x,y)$ , then the first partial derivatives of  with respect to $x$ and $y$ are the functions ${\frac {\partial z}{\partial x}}$ and ${\frac {\partial z}{\partial x}}$ , defined as shown.

${\frac {\partial z}{\partial x}}=\lim _{\Delta x\to 0}{\frac {f(x+\Delta x,y)-f(x,y)}{\Delta x}}$ ${\frac {\partial z}{\partial y}}=\lim _{\Delta y\to 0}{\frac {f(x,y+\Delta y)-f(x,y)}{\Delta y}}$ We can denote ${\frac {\partial z}{\partial x}}$ as $f_{x}(x,y)$ and ${\frac {\partial z}{\partial y}}$ as $f_{y}(x,y)$ Example: Find ${\frac {\partial z}{\partial x}}$ and ${\frac {\partial z}{\partial y}}$ of:

1) $z=f(x,y)=2x^{2}-4xy$ Solution:
${\frac {\partial z}{\partial x}}=4x^{2}-4y$ ${\frac {\partial z}{\partial y}}=-4x$ 2) $z=f(x,y)=x^{2}y^{3}$ Solution:
${\frac {\partial z}{\partial x}}=2xy^{3}$ ${\frac {\partial z}{\partial y}}=3x^{2}y^{2}$ 3) $z=f(x,y)=x^{2}e^{x^{2}y}$ Solution:
${\frac {\partial z}{\partial x}}=2xe^{x^{2}y}+x^{2}e^{x^{2}y}2xy$ (product rule +chain rule)
${\frac {\partial z}{\partial y}}=x^{2}e^{x^{2}y}(x^{2})=x^{4}e^{x^{2}y}$ 