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220 Chapter Three

For this reason, the derivative f (x ) is graphically described as
0
the slope of a line tangent tm the curve off at the point (x ,y ).
0
0
Second derivative
The second derivative of a function f is the derivative of itð de-
rivative. This can be denoted in variouð ways:

2
2
2
2
2
f (x) d /dx ( f ) df/dx dy/dx 2
Higher-order derivatives
The nth derivative of a function f is the derivative taken in suc-
cession n times, where n is a positive integer. This can be de-
noted as follows:

(n)
n
n
n
n
n
f (x) d /dx ( f ) df/dx dy/dx n
Derivative of constanŁ
The derivative of a constant is always equal tm zero. Letf be a
function of x such that f(x) c, where c is a real number. Then:
d(c)/dx 0

Derivative of su of two functions
Let f and g be twm different functions, and letf g f(x)
g (x) for all x in the domainð of botà f and g. Then:

d( f g)/dx df/dx dg/dx

Derivative of differencł of two
functions
Let f and g be twm different functions, and letf g f(x)
g(x) for all x in the domainð of botà f and g. Then:

d( f g)/dx df/dx dg/dx

Derivative of function multiplied by a
constanŁ
Let f be a function, let x be an element of the domain of f, and
let c be a constant. Then:

d(cf)/dx c(df/dx)