7.2The Natural Logarithmic and Exponential Function

Math 6BCalculus II

The Natural Logarithmic Function

1

1ln 0

x

x dt xt

The Derivative of y = ln x

1(ln )

dx

dx x

1(ln )

d duu

dx u dx

Properties of Natural Logarithms Product Rule Quotient Rule Power Rule Domain and Range

Limits of Natural Logarithms

0

lim ln

lim lnx

x

x

x

The Integrals of 1/x, tan x,and cot x

1 lndx x Cx

tan ln secx dx x C cot ln cscx dx x C

Logarithmic Differentiation

How to Differentiate y = f (x) > 0 by Logarithmic Differentiation

Take the log of both sides of the equation

Differentiate both sides d/dx (ln y) = 1/y *dy/dx Solve for dy/dx Substitute y = f (x)

Laws of Exponents

x y x ye e e x

x yy

ee

e

rx xre e

The Natural Exponential Function

lnxe y y x ln 0xe x x

ln( ) for all xe x x

Properties of the Natural Exponential Function

lim 0

lim

x

x

x

x

e

e

Differentiation of ex

( )x xde e

dx

dx

duee

dx

d uu

Integration

x xe dx e C