7.2the natural logarithmic and exponential function math 6b calculus ii
TRANSCRIPT
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