2017 Special Derivativesex , ax , ln (x) , log ax

AP Calculus

Derivative of - Analytically

REM: 0

1lim 1x

x

ex

xd e

dx

xe

ud edx

x xd e edx

Chain Rule:

Ex:

2 3x xy e

2sin ( )xy e

5xy e

ax , ln (x) , log ax

REM: IFFlog ( )a y x xy aTwo Properties from the Definition:

( )log xa a xlog ( )a yy a

ALSO: IFF

Two Properties from the Definition:

ln( )y x xy e

ln( )yy e ln xe x

Proof: ln(x)

ln( )d xdx

ln(x)

EX:

EX: NOTE:

2ln( 2 3)y x x

ln cos( )y x

ln(x)

EX: Find the second derivative.

2ln(1 )y x

Proof: ax

xd adx

ax Ex: 2( 2)5 xy

2 2ty t

Proof: loga(x)

log ( )ad xdx

loga(x)

3log ( )d xdx

24log ( 5 )y x x

ln( )

log ( )

u

u

a

d edx

d adx

d udx

d udx

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Logarithmic Differentiation

REM: Properties of Logarithms

2

ln abc

Rewrite using properties of Logs.

Logarithmic Differentiation

Set the function equal to y.

Take the natural log of both sides.Take the derivative of BOTH sides - f(y) and f(x) (implicitly)

Solve for dy/dx.

Then resub for y.

ln( ) ln( )

kt

kt

kt

d Cedt

y Ce

y Ce

Derivative :“Function raised to a Function Power”

sin( )xy x

Logarithmic Differentiation

2 2

3

( 1)

2 1

x xyx

Last Update

• 10/20/10

• Assignment: p. 178 # 1 – 31 odd 43, 45