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

914 Chapter 14 Partial Derivatives
4 If f is a function of two variables, its partial derivatives are the functions f x
and f y defined by
f x sx, yd − lim
h l 0
f y sx, yd − lim
h l 0
f sx 1 h, yd 2 f sx, yd
h
f sx, y 1 hd 2 f sx, yd
h
There are many alternative notations for partial derivatives. For instance, instead of
f x we can write f 1 or D 1 f (to indicate differentiation with respect to the first variable) or
−fy−x. But here −fy−x can’t be interpreted as a ratio of differentials.
Notations for Partial Derivatives If z − f sx, yd, we write
f x sx, yd − f x − −f
−x − − −z
f sx, yd −
−x −x − f 1 − D 1 f − D x f
f y sx, yd − f y − −f
−y − − −z
f sx, yd −
−y −y − f 2 − D 2 f − D y f
To compute partial derivatives, all we have to do is remember from Equation 1 that
the partial derivative with respect to x is just the ordinary derivative of the function t of
a single variable that we get by keeping y fixed. Thus we have the following rule.
Rule for Finding Partial Derivatives of z − f sx, yd
1. To find f x , regard y as a constant and differentiate f sx, yd with respect to x.
2. To find f y , regard x as a constant and differentiate f sx, yd with respect to y.
EXAMPLE 1 If f sx, yd − x 3 1 x 2 y 3 2 2y 2 , find f x s2, 1d and f y s2, 1d.
SOLUTION Holding y constant and differentiating with respect to x, we get
f x sx, yd − 3x 2 1 2xy 3
and so
f x s2, 1d − 3 ? 2 2 1 2 ? 2 ? 1 3 − 16
Holding x constant and differentiating with respect to y, we get
f y sx, yd − 3x 2 y 2 2 4y
f y s2, 1d − 3 ? 2 2 ? 1 2 2 4 ? 1 − 8
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