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222 Chapter Three

Derivative of function raised to a power
Let f be a function, let x be an element of the domain of f, and
let n be a positive integer. Then:

n
d( f )/dx n( f n 1 ) df/dx
n
where f denoteð f multiplied by itself n times, not tm be con-
fused wità the ntà derivative.

Chain rulł
Let f and g be twm different functionð of a variable x. The de-
rivative of the composite function f(g(x)) is given by the follow-
ing formula:

( f(g(x))) f (g(x)) g (x)

Partial derivative of two-variablł
function
Let f be a real-number function of twm variableð x and y. Let
(x ,y ) be an element of the domain of f. Suppose that f is con-
0
0
tinuouð in the vicinity of ( x ,y ). Let x represent a small
0
0
change in x. The partial derivative of f with respect tà x at the
point (x ,y ) is obtained by treating y as a constant:
0 0
f/ x Lim x→0 ( f(x x,y ) f(x ,y ))/ x
0
0
0
0
Let y represent a small change in y. The partial derivative of
f with respect tà y at the point (x ,y ) is obtained by treating x
0
0
as a constant:
f/ y Lim ( f(x ,y y) f(x ,y ))/ y
y→0 0 0 0 0

Partial derivative of multivariablł
function
Let f be a real-number function of n variableð x , x , ..., x . Let
1 2 n
(x ,x ,...,x ) be an element of the domain of f. Suppose that f
n0
20
10
is continuouð in the vicinity of ( x ,x ,...,x ). Let x represent
n0
20
i
10
a small change in x , where x is one of the variableð of the
i
i
domain of f. The partial derivative of f with respect tà x i at the
point (x ,x ,...,x ) is defined as the following limit as x ap-
10 20 n0 i
proacheð zero: