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Partial Differentiation
Suppose you want to forecast the weather this weekend in Los Angeles.
You construct a formula for the temperature as a function of several
environmental variables, each of which is not entirely predictable.
Now you would like to see how your weather forecast would change as
one particular environmental factor changes, holding all the other
factors constant. To do this investigation, you would use the concept
of a partial derivative.
Let the temperature $T$ depend on variables $x$ and $y$, $T =
f(x,y)$. The rate of change of $f$ with respect to $x$ (holding $y$
constant) is called the partial derivative of $f$ with respect to $x$
and is denoted by $f_{x}(x,y)$. Similarly, the rate of change of
$f$ with respect to $y$ is called the partial derivative of $f$ with respect to $y$
and is denoted by $f_{y}(x,y)$.
We define
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$$
f_{x}(x,y) = \lim_{h \rightarrow 0} \frac{f(x+h,y)-f(x,y)}{h} \phantom{.}
$$
$$
f_{y}(x,y) = \lim_{h \rightarrow 0} \frac{f(x,y+h)-f(x,y)}{h}.
$$
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Do you see the similarity beween these and the limit
definition of a function of one variable?
Example
$\qquad\qquad \begin{array}{rcl@{\qquad}rcl}
\displaystyle
\mathrm{Let~}f(x,y) & = & xy^{2} & & & \\
\mathrm{Then~}f_{x}(x,y) & = & \lim\limits_{h \rightarrow 0}
\frac{(x+h)y^{2}-xy^2}{h} & f_{y}(x,y) & = & \lim\limits_{h \rightarrow 0}
\frac{x(y+h)^{2}-xy^2}{h}\\
& = & \lim\limits_{h \rightarrow 0} \frac{hy^{2}}{h} & & = & \lim\limits_{h
\rightarrow 0} \frac{2xyh+ xh^{2}}{h} \\
& = & y^{2}. & & = & \lim\limits_{h \rightarrow 0} (2xy + xh) \\
& & & & = & 2xy.
\end{array}$
In practice, we use our knowledge of single-variable calculus to
compute partial derivatives. To calculate $f_{x}(x,y)$, you view $y$
as a constant and differentiate $f(x,y)$ with respect to $x$:
$$
f_{x}(x,y) =y^{2} \mathrm{~as~expected~since~} \frac{d}{dx}[x]= 1.
$$
Similarly,
$$
f_{y}(x,y) = 2xy \mathrm{~since~} \frac{d}{dy}\left[y^{2}\right]= 2y.
$$
More Examples
Notation
- Let $z = f(x,y)$.
The partial derivative $f_{x}(x,y)$ can also be written as
$$
\frac{\partial f}{\partial x}(x,y) \mathrm{~or~} \frac{\partial
z}{\partial x}.
$$
Similarly, $f_{y}(x,y)$ can also be written as
$$
\frac{\partial f}{\partial y}(x,y) \mathrm{~or~} \frac{\partial
z}{\partial y}.
$$
- The partial derivative $f_{y}(x,y)$ evaluated at the point
$(x_{0},y_{0})$ can be expressed in several ways:
$$
f_{x}(x_{0},y_{0}) \mathrm{,~} \left. \frac{\partial f}{\partial x}
\right|_{(x_{0},y_{0})} \mathrm{,~or~} \frac{\partial f}{\partial
x}(x_{0},y_{0}) .
$$
There are analogous expressions for $f_{y}(x_{0},y_{0})$.
Geometrical Meaning
Suppose the graph of $z = f(x,y)$ is the surface shown. Consider the
partial derivative of $f$ with respect to $x$ at a point
$(x_{0},y_{0})$.
Holding $y$ constant and varying $x$, we trace out a curve that is the
intersection of the surface with the vertical plane $y= y_{0}$.
The partial derivative $f_{x}(x_{0},y_{0})$ measures the change in $z$
per unit increase in $x$ along this curve. That is,
$f_{x}(x_{0},y_{0})$ is just the slope of the curve at
$(x_{0},y_{0})$. The geometrical interpretation of
$f_{y}(x_{0},y_{0})$ is analogous.
Notes
- Functions of More than Two Variables
For $g(x,y,z)$, the partial derivative $g_{x}(x,y,z)$ is calculated by
holding $y$ and $z$ constant and differentiating with respect to $x$.
The partial derivatives $g_{y}(x,y,z)$ and $g_{z}(x,y,z)$ are
calculated in an analagous manner.
- Higher-Order Partial Derivatives
For a function $f(x,y)$, the partial derivatives $\frac{\partial
f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are themselves
functions of $x$ and $y$, so we can take partial derivatives of them:
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$$
f_{xx}=
\frac{\partial}{\partial x} \left( \frac{\partial f}{\partial x}
\right) = \frac{\partial^{2} f}{\partial x^{2}} \qquad f_{xy} =
\frac{\partial}{\partial y} \left( \frac{\partial f}{\partial x}
\right) = \frac{\partial^{2} f}{\partial y \partial x} \phantom{.}
$$
$$
f_{yy} = \frac{\partial}{\partial y} \left( \frac{\partial f}{\partial y}
\right) = \frac{\partial^{2} f}{\partial y^{2}} \qquad
f_{yx} = \frac{\partial}{\partial x} \left( \frac{\partial f}{\partial y}
\right) = \frac{\partial^{2} f}{\partial x \partial y}.
$$
Higher-order partial derivatives (e.g. $f_{xxy}$) can also be
calculated. Using the subscript notation, the order of
differentiation is from left to right.
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$f_{xy}$ and $f_{yx}$ are called mixed second-order partial derivatives.
If $f$, $f_{x}$, $f_{y}$, $f_{xy}$, and $f_{yx}$ are
continuous on an open region, then $f_{xy} = f_{yx}$ at each point in
the region, so the order in which the differentiation is done does not
matter.
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Key Concepts
Consider a function $f(x,y)$.
| $$ f_{x}(x,y) $$
| $\quad = \quad$
| rate of change of $f$
with respect to $x$
| $\quad = \quad$
| $$ \lim_{h \rightarrow 0} \frac{f(x+h, y)- f(x,y)}{h} \phantom{.} $$
|
| $$ f_{y}(x,y) $$
| $\quad = \quad$
| rate of change of $f$
with respect to $y$
| $\quad = \quad$
| $$ \lim_{h \rightarrow 0} \frac{f(x, y+h)- f(x,y)}{h}. $$
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To calculate $f_{x}(x,y)$, differentiate $f$ with respect to $x$
holding $y$ constant. Similarly, to calculate $f_{y}(x,y)$,
differentiate $f$ with respect to $y$ holding $x$ constant.
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