The Mathematics of Financial Modelingand Investment Management

4-PrincipCalculus Page 111 Friday, March 12, 2004 12:39 PM

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Principles of Calculus

EXHIBIT 4.7 Commonly Used Derivatives

f(x)

xn

α

sin x

cos x

tan x

ln x

ex

log (f(x))

df

dx

nxn–1

axα–1

cos x

–sin x

1

cos^2 ()x

1

x

ex

f ′()x

fx()

Domain of P

R, x ≠0 if n < 0

x > 0

R

R

• ---π+ n---π <<x π---+ nπ ---
  2 2 2 2

x > 0

R

f(x) ≠ 0

Note: Where R denotes real numbers.

Given a function y = f(x), its increments ∆f = f(x + ∆x) – f(x) can be

approximated by

∆ fx()= f ′()x∆x

The quality of this approximation depends on the function itself.

HIGHER ORDER DERIVATIVES

Suppose that a function f(x) is differentiable in an interval D and its

derivative is given by

f′()x = df x()

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dx

The derivative might in turn be differentiable. The derivative of a deriv-

ative of a function is called a second-order derivative and is denoted by