CHAPTER 14
DIFFERENTIATING FUNCTIONS OF
SEVERAL VARIABLES
I For a function of one variable, y = f (x), the derivative
dy /dx = f 0 (x) gives the rate of change of y with respect to
x.
I For a function of one variable, y = f (x), the derivative
dy /dx = f 0 (x) gives the rate of change of y with respect to
x.
I For a function of two variables, z = f (x, y), there is no such
thing as the rate of change, since x and y can each vary
while the other is held fixed or both can vary at once.
I For a function of one variable, y = f (x), the derivative
dy /dx = f 0 (x) gives the rate of change of y with respect to
x.
I For a function of two variables, z = f (x, y), there is no such
thing as the rate of change, since x and y can each vary
while the other is held fixed or both can vary at once.
I However, we can consider the rate of change with respect
to each one of the independent variables.
I For a function of one variable, y = f (x), the derivative
dy /dx = f 0 (x) gives the rate of change of y with respect to
x.
I For a function of two variables, z = f (x, y), there is no such
thing as the rate of change, since x and y can each vary
while the other is held fixed or both can vary at once.
I However, we can consider the rate of change with respect
to each one of the independent variables.
I This chapter introduces these partial derivatives and
several ways they can be used to get a complete picture of
the way the function varies.
14.1 The Partial Derivative
The derivative of a one-variable function measures its rate of
change. In this section we see how a two variable function
f (x, y ) has two rates of change: one as x changes (with y held
constant) and one as y changes (with x held constant).
Rate of Change of Monthly Payments: Student Loan
The following table shows monthly payments P of a student
loan as a function of two variables: interest rate and loan
amount. Estimate the rate of change of P for a $7,000 loan at
5% with respect to each variable.
Rate of Change of Monthly Payments: Car Loan
The following contour diagram shows monthly payments P of a
car loan as a function of two variables: interest rate and loan
amount. Estimate the rate of change of P for a $6,000 loan at
7.5% with respect to each variable.
Definition of the Partial Derivative
We study the influence of x and y separately on the value of
the function f (x, y) by holding one fixed and letting the other
vary. This leads to the following definitions.
Definition of the Partial Derivative
We study the influence of x and y separately on the value of
the function f (x, y) by holding one fixed and letting the other
vary. This leads to the following definitions.
For all points at which the limits exist, we define the partial
derivative fx (a, b) of f (x, y ) with respect to x at the point (a, b)
by
f (a + h, b) − f (a, b)
fx (a, b) = lim .
h→0 h
Definition of the Partial Derivative
We study the influence of x and y separately on the value of
the function f (x, y) by holding one fixed and letting the other
vary. This leads to the following definitions.
For all points at which the limits exist, we define the partial
derivative fx (a, b) of f (x, y ) with respect to x at the point (a, b)
by
f (a + h, b) − f (a, b)
fx (a, b) = lim .
h→0 h
Similarly, we define the partial derivative fy (a, b) of f (x, y ) with
respect to y at the point (a, b) by
f (a, b + h) − f (a, b)
fy (a, b) = lim .
h→0 h
Definition of the Partial Derivative
We study the influence of x and y separately on the value of
the function f (x, y) by holding one fixed and letting the other
vary. This leads to the following definitions.
For all points at which the limits exist, we define the partial
derivative fx (a, b) of f (x, y ) with respect to x at the point (a, b)
by
f (a + h, b) − f (a, b)
fx (a, b) = lim .
h→0 h
Similarly, we define the partial derivative fy (a, b) of f (x, y ) with
respect to y at the point (a, b) by
f (a, b + h) − f (a, b)
fy (a, b) = lim .
h→0 h
If we let a and b vary, we have the partial derivative functions
fx (x, y ) and fy (x, y).
Alternative Notation for Partial Derivatives
If z = f (x, y ), we can write
∂z ∂z
fx (x, y) = and fy (x, y ) =
∂x ∂y
Alternative Notation for Partial Derivatives
If z = f (x, y ), we can write
∂z ∂z
fx (x, y) = and fy (x, y ) =
∂x ∂y
∂z ∂z
fx (a, b) = and fy (a, b) =
∂x (a,b) ∂y (a,b)
Alternative Notation for Partial Derivatives
If z = f (x, y ), we can write
∂z ∂z
fx (x, y) = and fy (x, y ) =
∂x ∂y
∂z ∂z
fx (a, b) = and fy (a, b) =
∂x (a,b) ∂y (a,b)
I We use the symbol ∂ to distinguish partial derivatives from
ordinary derivatives.
Alternative Notation for Partial Derivatives
If z = f (x, y ), we can write
∂z ∂z
fx (x, y) = and fy (x, y ) =
∂x ∂y
∂z ∂z
fx (a, b) = and fy (a, b) =
∂x (a,b) ∂y (a,b)
I We use the symbol ∂ to distinguish partial derivatives from
ordinary derivatives.
I In cases where the independent variables have names
different from x and y, we adjust the notation accordingly.
Alternative Notation for Partial Derivatives
If z = f (x, y ), we can write
∂z ∂z
fx (x, y) = and fy (x, y ) =
∂x ∂y
∂z ∂z
fx (a, b) = and fy (a, b) =
∂x (a,b) ∂y (a,b)
I We use the symbol ∂ to distinguish partial derivatives from
ordinary derivatives.
I In cases where the independent variables have names
different from x and y, we adjust the notation accordingly.
I For example, the partial derivatives of f (u, v ) are denoted
by fu and fv .
Visualizing Partial Derivatives on a Graph
The ordinary
derivative of a one-variable
function is the slope of its graph.
Visualizing Partial Derivatives on a Graph
The ordinary
derivative of a one-variable
function is the slope of its graph.
How do we visualize
the partial derivative fx (a, b)?
Visualizing Partial Derivatives on a Graph
The ordinary
derivative of a one-variable
function is the slope of its graph.
How do we visualize
the partial derivative fx (a, b)?
The graph of the one-variable
function f (x, b) is the
curve where the vertical plane
y = b cuts the graph of f (x, y).
Visualizing Partial Derivatives on a Graph
The ordinary
derivative of a one-variable
function is the slope of its graph.
How do we visualize
the partial derivative fx (a, b)?
The graph of the one-variable
function f (x, b) is the
curve where the vertical plane
y = b cuts the graph of f (x, y).
Thus, fx (a, b) is the slope of
the tangent line to this curve at
x = a.
Visualizing Partial Derivatives on a Graph
Similarly, the graph
of the function f (a, y ) is the
curve where the vertical plane
x = a cuts the graph of f , and
the partial derivative fy (a, b) is
the slope of this curve at y = b.
Using Units to Interpret Partial Derivatives
The meaning of a partial derivative can often be explained
using units.
Using Units to Interpret Partial Derivatives
The meaning of a partial derivative can often be explained
using units.
Example
The sales of a product, S = f (p, a), is a function of the price, p,
of the product (in dollars per unit) and the amount, a, spent on
advertising (in thousands of dollars).
Using Units to Interpret Partial Derivatives
The meaning of a partial derivative can often be explained
using units.
Example
The sales of a product, S = f (p, a), is a function of the price, p,
of the product (in dollars per unit) and the amount, a, spent on
advertising (in thousands of dollars).
(a) Do you expect fp to be positive or negative? Why?
Using Units to Interpret Partial Derivatives
The meaning of a partial derivative can often be explained
using units.
Example
The sales of a product, S = f (p, a), is a function of the price, p,
of the product (in dollars per unit) and the amount, a, spent on
advertising (in thousands of dollars).
(a) Do you expect fp to be positive or negative? Why?
(b) Using the units for p and a, interpret the statement
fa (8, 12) = 150 in terms of sales.
Example
In a heated room, H(x, t) gives the temperature, in ◦ C, as a
function of t, time after the heater has been turned on in
minutes, and x, the distance away from the heater in meters.
Example
In a heated room, H(x, t) gives the temperature, in ◦ C, as a
function of t, time after the heater has been turned on in
minutes, and x, the distance away from the heater in meters.
1. Describe in practical terms what Hx (x, t) and Ht (x, t)
stands for.
2. Estimate Hx (10, 20) and explain your answer in practical
terms.
3. Estimate Ht (10, 20) and explain your answer in practical
terms.
4. Do you expect Hx (x, t) and Ht (x, t) to be positive
everywhere, negative everywhere, or to change signs?
Explain your answer.
Thank You
