1.9: Partial Derivatives
Definition of a Partial Derivative
Let \(f(x,y)\) be a function of two variables. Then we define the partial derivatives as:
Definition: Partial Derivative
\[ f_x = \dfrac{\partial f}{\partial x} = \lim_{h\to{0}} \dfrac{f(x+h,y)f(x,y)}{h} \]
\[ f_y = \dfrac{\partial f}{\partial y} = \lim_{h\to{0}} \dfrac{f(x,y+h)f(x,y)}{h} \]
if these limits exist.
Algebraically, we can think of the partial derivative of a function with respect to \(x\) as the derivative of the function with \(y\) held constant. Geometrically, the derivative with respect to \(x\) at a point \(P\) represents the slope of the curve that passes through \(P\) whose projection onto the \(xy\) plane is a horizontal line (if you travel due East, how steep are you climbing?)
Example 1
Let
\[ f(x,y) = 2x + 3y \]
then
\[\begin{align} \dfrac{\partial f}{ \partial } &= \lim_{h\to{0}}\dfrac{(2(x+h)+3y)  (2x+3y)}{h} \\ &= \lim_{h\to{0}} \dfrac{2x+2h+3y2x3y}{h} \\ &= \lim_{h\to{0}} \dfrac{2h}{h} =2 . \end{align}\]
We also use the notation \(f_x\) and \(f_y\) for the partial derivatives with respect to \(x\) and \(y\) respectively.
Find \(f_y\) for the function from the example above.
Finding Partial Derivatives the Easy Way
Since a partial derivative with respect to \(x\) is a derivative with the rest of the variables held constant, we can find the partial derivative by taking the regular derivative considering the rest of the variables as constants.
Example 2
Let
\[ f(x,y) = 3xy^2  2x^2y \]
then
\[ f_x = 3y^2  4xy\]
and
\[ f_y = 6xy  2x^2. \]
Exercises
Find both partial derivatives for

\(f(x,y) = xy \sin x \)

\( f(x,y) = \dfrac{ x + y}{ x  y}\).
Higher Order Partials
Just as with function of one variable, we can define second derivatives for functions of two variables. For functions of two variables, we have four types: \( f_{xx}\), \(f_{xy}\), \(f_{yx}\), and \(f_{yy}\).
Example 3
Let
\[f(x,y) = ye^x \]
then
\[f_x = ye^x \]
and
\[f_y=e^x. \]
Now taking the partials of each of these we get:
\[f_{xx}=ye^x \;\;\; f_{xy}=e^x \;\;\; \text{and} \;\;\; f_{yy}=0 . \]
Notice that
\[ f_{x,y} = f_{yx}.\]
Theorem
Let \(f(x,y)\) be a function with continuous second order derivatives, then
\[f_{xy} = f_{yx}. \]
Functions of More Than Two Variables
Suppose that
\[ f(x,y,z) = xy  2yz \]
is a function of three variables, then we can define the partial derivatives in much the same way as we defined the partial derivatives for three variables.
We have
\[f_x=y \;\;\; f_y=x2z \;\;\; \text{and} \;\;\; f_z=2y . \]
Example 4: The Heat Equation
Suppose that a building has a door open during a snowy day. It can be shown that the equation
\[ H_t = c^2H_{xx} \]
models this situation where \(H\) is the heat of the room at the point \(x\) feet away from the door at time \(t\). Show that
\[ H = e^{t} \cos(\frac{x}{c}) \]
satisfies this differential equation.
Solution
We have
\[H_t = e^{t} \cos (\dfrac{x}{c}) \]
\[H_x = \dfrac{1}{c} e^{t} \sin(\frac{x}{c})\]
\[H_{xx} = \dfrac{1}{c^2} e^{t} \cos(\dfrac{x}{c}) .\]
So that
\[c^2 H_{xx}= e^{t} \cos (\dfrac{x}{c}) . \]
And the result follows.
Contributors
 Larry Green (Lake Tahoe Community College)
Integrated by Justin Marshall.