James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

918 Chapter 14 Partial Derivatives
respect to x when y and z are held fixed. But we can’t interpret it geometrically because
the graph of f lies in four-dimensional space.
In general, if u is a function of n variables, u − f sx 1 , x 2 , . . . , x n d, its partial deriva tive
with respect to the ith variable x i is
−u f sx 1 , . . . , x i21 , x i 1 h, x i11 , . . . , x n d 2 f sx 1 , . . . , x i , . . . , x n d
− lim
−x i h l 0 h
and we also write
−u
− −f − f xi − f i − D i f
−x i −x i
EXAMPLE 6 Find f x , f y , and f z if f sx, y, zd − e x y ln z.
SOLUTION Holding y and z constant and differentiating with respect to x, we have
f x − ye x y ln z
Similarly, f y − xe x y ln z and f z − e xy
z
■
Higher Derivatives
If f is a function of two variables, then its partial derivatives f x and f y are also functions
of two variables, so we can consider their partial derivatives s f x d x , s f x d y , s f y d x , and s f y d y ,
which are called the second partial derivatives of f. If z − f sx, yd, we use the following
notation:
s f x d x − f xx − f 11 − S − −f −
−x −xD −2 f
−x 2
s f x d y − f xy − f 12 − S − −f −
−y −xD
s f y d x − f yx − f 21 − S − −f −
−x −yD
s f y d y − f yy − f 22 − S − −f −
−y −yD −2 f
−y 2
− −2 z
−x 2
−2 f
−y −x −
−2 f
−x −y −
− −2 z
−y 2
−2 z
−y −x
−2 z
−x −y
Thus the notation f x y (or − 2 fy−y −x) means that we first differentiate with respect to x and
then with respect to y, whereas in computing f yx the order is reversed.
EXAMPLE 7 Find the second partial derivatives of
SOLUTION In Example 1 we found that
Therefore
f x sx, yd − 3x 2 1 2xy 3
f sx, yd − x 3 1 x 2 y 3 2 2y 2
f y sx, yd − 3x 2 y 2 2 4y
f xx − − −x s3x 2 1 2xy 3 d − 6x 1 2y 3 f xy − − −y s3x 2 1 2xy 3 d − 6xy 2
f yx − − −x s3x 2 y 2 2 4yd − 6xy 2 f yy − − −y s3x 2 y 2 2 4yd − 6x 2 y 2 4 ■
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