Partial Derivatives - University of Pennsylvania

10/10/2012

Math 114 ? Rimmer 14.3 Partial Derivatives

14.3

Partial Derivatives

In this section, we will learn about: Multivariable Derivatives

Math 114 ? Rimmer 14.3 Partial Derivatives

fx (x, y) =

lim

h0

f

(x + h, y)-

h

f

(x, y)

Partial derivative of f with respect to x as a function itself

Regard y as a constant and differentiate f (x, y) with respect to x

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Math 114 ? Rimmer 14.3 Partial Derivatives

f y (x,

y) =

lim

h0

f

(x,

y

+ h)-

h

f

(x,

y)

Partial derivative of f with respect to y as a function itself

Regard x as a constant and differentiate f (x, y) with respect to y

Math 114 ? Rimmer 14.3 Partial Derivatives

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Math 114 ? Rimmer 14.3 Partial Derivatives

Math 114 ? Rimmer 14.3 Partial Derivatives

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Notation:

Math 114 ? Rimmer 14.3 Partial Derivatives

fx (x, y) =

fx

=

f x

=

x

f (x, y) =

z x

=

Dx f

fxy (x, y) =

f xy

=

2 f yx

The derivative with respect to x first,

then the derivative with respect to y of that.

fxx (x,

y) =

f xx

=

2 f x 2

The derivative with respect to x first,

then the derivative with respect to x of that.

Math 114 ? Rimmer 14.3 Partial Derivatives

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Clairaut's Theorem

fxy (x, y) = f yx (x, y)

Math 114 ? Rimmer 14.3 Partial Derivatives

Mixed partials are equal.

3 Classical Partial Differential Equations (PDEs)

ut = kuxx

Heat Equation

a2uxx = utt

Wave Equation

uxx + uyy = 0

Laplace's Equation

f (x, y) = 3x2 y + y3 - 3x2 - 3y2 + 2

Math 114 ? Rimmer 14.3 Partial Derivatives

fx = 6xy - 6x f y = 3x2 + 3y2 - 6 y

fxx = 6 y - 6 f yy = 6 y - 6

fxy = 6x

f yx = 6x

g

(

x,

y)

=

x2

y3

+

ln

x

y

gx

=

2xy3

+

1

x y

1 y

=

2xy3

+

1 x

g xx

=

2 y3

-

1 x2

gy

= 3x2 y2

+

1

x y

-x y2

=

3x2 y2

-

1 y

g yy

=

6x2 y

+

1 y2

gxy = g yx = 6xy2

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