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Calc 3 Lecture Notes Section 12.3 of 6

Section 12.3: Partial Derivatives

Big idea: The notion of the derivative of a single-variable function can be extended to a

multivariate function if a derivative is taken with respect to one variable while holding thevalue(s) of the other variable(s) constant. This is called a partial derivative.

Big skill: You should be able to compute first-order and higher-order partial derivatives.

The derivative f ′(a) of a univariate function f ( x) tells us the slope of the tangent line to

the curve y = f ( x) at the point (a, f (a)). The partial derivative ( ), f

a b x

∂

∂tells us the slope of the

tangent line to the surface ( ), z f x y= at the point (a, b, f (a, b)) in the plane y = b. The partial

derivative ( ), f

a b y

∂

∂tells us the slope of the tangent line to the surface ( ), z f x y= at the point (a,

b, f (a, b)) in the plane x = a.

A graph of ( ) 23 y f x x= = − and the

tangent line to the curve at (1, 2). Note

that ( ) 2 f x x′ = − and the equation of the

tangent line is 2 4 y x= − + .

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A graph of ( ) 2 2, 4 z f x y x y= = − − and the

plane y = 1. The plane intersects the surface

along a curve specified by

2

31

z x y

= − =

. The

tangent line to the surface at (1, 1, 2) in the

plane y = 1 is also shown. Note that in this

plane, ( ) 2 z x x′ = − and the equation of the

tangent line is 1

2 4

x t

y

z t

=

= = − +

For the function ( ) 2 2, 4 f x y x y= − − ,

( ) 2 2, 4 0 2 0 2 f x y x y x x

x x∂ ∂ = − − = − − = − ∂ ∂

.

A graph of ( ) 2 2, 4 z f x y x y= = − − and the

plane x = 1. The plane intersects the surface

along a curve specified by

2

31

z y x

= − =

. The

tangent line to the surface at (1, 1, 2) in the

plane x = 1 is also shown. Note that in this

plane, ( ) 2 z y y′ = − and the equation of the

tangent line is

1

2 4

x

y t

z t

=

= = − +

For the function ( ) 2 2, 4 f x y x y= − − ,

( ) 2 2, 4 0 0 2 2 f x y x y y y

y y∂ ∂ = − − = − − = − ∂ ∂

.

Definition 3.1: Partial Derivative

The partial derivative of f ( x , y) with respect to x , written as f

x

∂

∂, is defined by

( )( ) ( )

0

, ,, lim

h

f x h y f x y f x y

x h→

+ −∂=

∂

for any values of x and y for which the limit exists.

The partial derivative of f ( x , y) with respect to y, written as f

y

∂

∂, is defined by

( )( ) ( )

0

, ,, lim

h

f x y h f x y f x y

y h→

+ −∂=

∂

for any values of x and y for which the limit exists.

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Note: To compute a partial derivative in practice, just treat all independent variables as constants

except for the variable with respect to which the derivative is being taken.

Various notations for partial derivatives of ( ), z f x y= :

( ) ( ) ( ) ( ), , , , x

f z x y f x y x y f x y

x x x

∂ ∂ ∂= = = ∂ ∂ ∂

(Likewise for partial derivatives with respect to y)

The expression f

x

∂

∂is a partial differential operator; it indicates to take the partial derivative with

respect to x of whatever expression follows it.

Practice:1. Compute ( ),

f x y

x

∂

∂and ( ),

f x y

y

∂

∂for ( ) 2, 5 2 f x y x y= − using the limit definition of the

derivative.

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2. Compute ( ), f

x y x

∂

∂for ( ) 2 2 2 2

, 3 4 2 7 9 f x y x y x y xy= + − + −

3. Compute ( ), f

x y y

∂

∂for ( ) ( ) ( ), cos sin xy f x y x y e= −

Higher order partial derivatives are partial derivatives of partial derivatives.

For a function of two variables, there are four different second-order partial derivatives:

• The partial derivative with respect to x of f

x

∂

∂is

f

x x

∂ ∂ ∂ ∂

.

Alternative notations:

2

2 xx

f f f

x x x

∂ ∂ ∂ = = ∂ ∂ ∂

• The partial derivative with respect to y of f

y

∂

∂is

f

y y

∂ ∂ ∂ ∂

.

Alternative notations:

2

2 yy f f f

y y y ∂ ∂ ∂= = ∂ ∂ ∂

• The partial derivative with respect to x of f

y

∂

∂is

f

x y

∂ ∂ ∂ ∂

.

Alternative notations: ( )2

y yx x

f f f f

x y x y

∂ ∂ ∂= = = ∂ ∂ ∂ ∂

. This is a mixed second-order

partial derivative.

• The partial derivative with respect to y of f

x

∂

∂is

f

y x

∂ ∂ ∂ ∂

.

• Alternative notations: ( )2

x xy y

f f f f y x y x∂ ∂ ∂ = = = ∂ ∂ ∂ ∂

. This is also a mixed second-order

partial derivative.

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Practice:

1. Compute all four second-order partial derivatives for

( ) 2 2 2 2, 3 4 2 7 9 f x y x y x y xy= + − + − .

2. Compute all four second-order partial derivatives for ( ) ( ) ( ), cos sin xy f x y x y e= − .

Notice that the mixed second-order partial derivatives are equal in both cases above… this isusually the case …

Theorem 3.1 (Equality of Mixed Second-Order Partial Derivatives)

If f xy( x, y) and f yx( x, y) are continuous on an open set containing (a, b), then f xy( x, y) = f yx( x, y).

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Practice:

1. Compute xyx f and xyxy f for ( ) ( ) ( ), cos sin xy f x y x y e= −

2. Compute x f , xy f , and xyz f for ( ) 2 3 4, , f x y z xyz x y z = +

3. The sag S in a beam of length L, width w, and height h is given by ( )4

3, ,

LS L w h c

wh= .

Write all three first-order partial derivatives in terms of S and one other variable to

determine which variable has the greatest proportional effect on the sag.

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