Partial Derivatives
Textbook Reference
| Primary source | OpenStax Calculus Volume 3, Section 4.3: “Partial Derivatives” |
| Direct link | https://openstax.org/books/calculus-volume-3/pages/4-3-partial-derivatives |
| Textbook used in class | Stewart, Calculus, Section 14.3: “Partial Derivatives” (Examples 1, 2, 3) |
Quick Reference
Definition: $$f_x(a,b) = \lim_{h\to 0}\frac{f(a+h,b)-f(a,b)}{h}, \qquad f_y(a,b) = \lim_{h\to 0}\frac{f(a,b+h)-f(a,b)}{h}.$$
Computing $f_x$: treat $y$ as a constant and differentiate with respect to $x$.
Computing $f_y$: treat $x$ as a constant and differentiate with respect to $y$.
Geometric meaning: $f_x(a,b)$ is the slope of the curve obtained by intersecting the surface $z = f(x,y)$ with the plane $y = b$.
Motivation
Suppose the temperature $T(x,y)$ at a point on a metal plate depends on two coordinates. If you walk in the $x$-direction, the rate of change of temperature is $\partial T/\partial x$. If you walk in the $y$-direction, it is $\partial T/\partial y$. A partial derivative measures the instantaneous rate of change in one coordinate direction while all other coordinates are held fixed.
This is the natural extension of the single-variable derivative to functions of several variables. The computation rule is almost the same: just treat all variables other than the one being differentiated as constants.
Key Concepts
1. The Definition
The partial derivative $f_x(a,b)$ is defined exactly like the ordinary derivative, but we only let $x$ vary (holding $y = b$ fixed): $$f_x(a,b) = \lim_{h\to 0}\frac{f(a+h,b)-f(a,b)}{h}.$$
This is a one-variable limit -- the same formula as the single-variable derivative, applied to the one-variable function $g(x) = f(x, b)$.
2. The Computation Rule
For most functions, there is no need to use the limit definition. Instead:
- To find $f_x$: treat $y$ as a constant and apply all the single-variable differentiation rules to $f$ viewed as a function of $x$.
- To find $f_y$: treat $x$ as a constant and differentiate with respect to $y$.
3. Geometric Picture
The surface $z = f(x,y)$ can be sliced by horizontal planes. The slice at $y = b$ is the curve $z = f(x,b)$, called a trace. The partial derivative $f_x(a,b)$ is the slope of this trace at $x = a$.
Worked Example
Find $f_x$ and $f_y$ for $f(x,y) = x^3 y + 5x^2 y^3 - xy^2$. (Stewart 14.3, Example 1.)
Finding $f_x$ (treat $y$ as a constant): $$f_x = 3x^2 y + 10x y^3 - y^2.$$
Finding $f_y$ (treat $x$ as a constant): $$f_y = x^3 + 15x^2 y^2 - 2xy.$$
Check at $(1,1)$: $f_x(1,1) = 3 + 10 - 1 = 12$, $f_y(1,1) = 1 + 15 - 2 = 14$.
Example 2. Find $f_x$ and $f_y$ for $f(x,y) = e^{xy}\sin(xy)$.
$f_x$: use the product rule, treating $y$ as a constant. $$f_x = ye^{xy}\sin(xy) + e^{xy}\cos(xy)\cdot y = ye^{xy}(\sin(xy) + \cos(xy)).$$
$f_y$: same structure, but differentiate with respect to $y$. $$f_y = xe^{xy}(\sin(xy) + \cos(xy)).$$
$\partial f/\partial x$ means differentiate $f$ with respect to both $x$ and $y$, then pick one. A partial derivative treats ALL variables except the one being differentiated as constants. When computing $f_x$ for $f = x^2 y^3$, the factor $y^3$ is a constant multiplier, so $f_x = 2x \cdot y^3 = 2xy^3$. Students sometimes try to differentiate $y^3$ and get an extra $3y^2$ factor. The rule is to hold $y$ completely fixed.
Leveled Practice
Problem 1. Find $f_x$ and $f_y$ for $f(x,y) = x^2\ln(y) + y\cos(x)$.
Show answer
$f_x = 2x\ln(y) - y\sin(x)$.
$f_y = x^2/y + \cos(x)$.