Download original notes
PAGE 1

15.3 Partial Derivatives

if \( y = f(x) \) the rate of change of \( y \) with respect to \( x \) is

\[ \frac{dy}{dx} = f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]

this tells us how the changing \( x \) affects \( y \)

if \( z = f(x, y) \) now \( z \) is affected by both \( x \) and \( y \) each of them affecting \( z \).

often we need to know how \( x \) and \( y \) individually affect \( z \).

example: wind chill factor

\[ W = 35.74 + 0.6215 T - 35.75 v^{0.16} + 0.4275 T v^{0.16} \quad (^{\circ}F) \]
  • \( T \): air temp \( (^{\circ}F) \)
  • \( V \): wind speed (mph)

current:

  • \( T = 8 \)
  • \( V = 10 \)
  • \( W = -6 \)
PAGE 2

\( z = f(x, y) \)

the partial derivative of \( f \) with respect to \( x \) is

\[ \frac{\partial f}{\partial x} = f_x = \lim_{h \to 0} \frac{f(x+h, y) - f(x, y)}{h} \]

weird "d": \( \partial \)

changing variable: \( x \)

\( x \) changes while \( y \) is held constant

the partial derivative of \( f \) with respect to \( y \) is

\[ \frac{\partial f}{\partial y} = f_y = \lim_{h \to 0} \frac{f(x, y+h) - f(x, y)}{h} \]

\( y \) changing

\( x \) constant

In practice, we use the differentiation rules we know but pretend the non changing variable is constant.

PAGE 3

Example: Partial Derivatives of a Polynomial

Given the function:

\[ f(x, y) = x^2 + y^2 + xy \]

Partial Derivative with respect to \( x \)

\( x \) is the variable, so \( y \) is treated as a constant.

\[ \frac{\partial f}{\partial x} = f_x = \frac{\partial}{\partial x} (x^2 + y^2 + xy) = 2x + 0 + y = \boxed{2x + y} \]

Partial Derivative with respect to \( y \)

\( y \) is the variable, so \( x \) is treated as a constant.

\[ \frac{\partial f}{\partial y} = f_y = \frac{\partial}{\partial y} (x^2 + y^2 + xy) = 0 + 2y + x = \boxed{x + 2y} \]
PAGE 4

Example: Partial Derivatives with Trigonometric Functions

Given the function:

\[ z = f(x, y) = x^3 + \tan(xy) \]

Partial Derivative with respect to \( x \)

\( y \) is constant.

\[ \begin{aligned} \frac{\partial f}{\partial x} = f_x = z_x &= \frac{\partial}{\partial x}(x^3) + \frac{\partial}{\partial x}(\tan(xy)) \\ &= 3x^2 + \sec^2(xy) \cdot \frac{\partial}{\partial x}(xy) \quad \text{(chain rule)} \\ &= \boxed{3x^2 + y \sec^2(xy)} \end{aligned} \]

Partial Derivative with respect to \( y \)

\( x \) is constant.

\[ \begin{aligned} \frac{\partial f}{\partial y} = f_y = z_y &= \frac{\partial}{\partial y}(x^3) + \frac{\partial}{\partial y}(\tan(xy)) \\ &= 0 + \sec^2(xy) \cdot \frac{\partial}{\partial y}(xy) \\ &= \boxed{x \sec^2(xy)} \end{aligned} \]
PAGE 5

Wind Chill Chart Analysis

A grid showing wind chill values for temperatures from 40 to -45°F and wind speeds from calm to 60 mph.

Wind Chill (°F) = \( 35.74 + 0.6215T - 35.75(V^{0.16}) + 0.4275T(V^{0.16}) \)
Where \( T \) = Air Temperature (°F) and \( V \) = Wind Speed (mph)

\( V \) is const

\[ \frac{\partial w}{\partial T} = 0.6215 + 0.4275 V^{0.16} \]

rate of change of wind chill as temp changes w/ \( V \) held constant. Follow a particular row.

\[ \frac{\partial w}{\partial V} = \dots \]

rate of change of \( w \) as wind speed varies. Follow a particular column.

PAGE 6

Example: Partial Derivatives

\[ f(x, y) = e^x \sin(y) \]

First-order partial derivatives

\[ \frac{\partial f}{\partial x} = f_x = e^x \sin(y) \]\[ \frac{\partial f}{\partial y} = f_y = e^x \cos(y) \]

Second-order partial derivatives

\[ \frac{\partial}{\partial x} \left( \frac{\partial f}{\partial x} \right) = \frac{\partial^2 f}{\partial x^2} = f_{xx} = \frac{\partial}{\partial x} (e^x \sin y) = e^x \sin(y) \]
\[ \frac{\partial}{\partial y} \left( \frac{\partial f}{\partial y} \right) = \frac{\partial^2 f}{\partial y^2} = f_{yy} = \frac{\partial}{\partial y} (e^x \cos y) = -e^x \sin(y) \]
\[ \frac{\partial}{\partial y} \left( \frac{\partial f}{\partial x} \right) = \frac{\partial^2 f}{\partial y \partial x} = f_{xy} = \frac{\partial}{\partial y} (e^x \sin y) = e^x \cos(y) \]

Note: order is "backwards" for Leibniz notation, but order is right for subscript notation.

\[ \frac{\partial}{\partial x} \left( \frac{\partial f}{\partial y} \right) = \frac{\partial^2 f}{\partial x \partial y} = f_{yx} = \frac{\partial}{\partial x} (e^x \cos y) = e^x \cos(y) \]

"mixed partials"

Mixed partials are equal if the four partial derivatives are continuous at the point.

PAGE 7

Example: Partial Derivatives

Given the function:

\[ f(x,y) = e^{x^2y} \]

The first-order partial derivatives are:

\[ \frac{\partial f}{\partial x} = f_x = 2xy e^{x^2y} \]\[ \frac{\partial f}{\partial y} = f_y = x^2 e^{x^2y} \]

Second-Order Partial Derivative with respect to x

\[ \frac{\partial^2 f}{\partial x^2} = f_{xx} = \frac{\partial}{\partial x} \left( \frac{\partial f}{\partial x} \right) = \frac{\partial}{\partial x} \left( (2xy) e^{x^2y} \right) \]
product of functions of x and y → product rule
\[ = (2xy) \cdot \frac{\partial}{\partial x} (e^{x^2y}) + (e^{x^2y}) \cdot \frac{\partial}{\partial x} (2xy) \]\[ = (2xy) \cdot e^{x^2y} \cdot 2xy + e^{x^2y} \cdot 2y = 4x^2y^2 e^{x^2y} + 2y e^{x^2y} \]

Mixed Partial Derivative

\[ f_{xy} = \frac{\partial}{\partial y} (f_x) = \frac{\partial}{\partial y} \left( (2xy) e^{x^2y} \right) \]\[ = (2xy) \cdot \frac{\partial}{\partial y} (e^{x^2y}) + (e^{x^2y}) \cdot \frac{\partial}{\partial y} (2xy) \]\[ = (2xy) \cdot e^{x^2y} \cdot (x^2) + (e^{x^2y}) \cdot (2x) \]\[ = 2x^3y e^{x^2y} + 2x e^{x^2y} = f_{yx} \]
PAGE 8

Example: Higher-Order Partial Derivatives

Given the function:

\[ f(x,y,z) = xyz \]

Find \( f_{xyz} \) and \( f_{yxz} \).

First-Order Partials

\[ f_x = yz \]\[ f_y = xz \]\[ f_z = xy \]

Second and Third-Order Partials

\[ f_{xy} = z \]
\[ f_{yx} = z \]
\[ f_{xyz} = 1 \]
\[ f_{yxz} = 1 \]
mixed partials are the same

Additional Calculation

\[ f_{zzz} = ? \]\[ f_{zz} = 0 = \frac{\partial}{\partial z} (xy) \]\[ \frac{\partial}{\partial z} (0) = 0 \]
PAGE 9

Please submit your reflections by using the CourseMIRROR App

If you are having a problem with CourseMIRROR, please send an email to coursemirror.development@gmail.com