John Napier was a Scottish scholar who is best known for his invention of logarithms, but other mathematical contributions include a mnemonic for formulas used in solving spherical triangles and two formulas known as Napier's analogies.

John Napier 1550 to 1617

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ln

Recall, if ( ) ln and ( ) then ( ( )) ln( )and ( ( )) (ln ) .

and are inverse functions of each other.

x

x

x

f x x g x ef g x e x

g f x g x e xf g

We would like to find its derivative.

Consider the exponential function , 0 and 1.xy b b b

So xy b

xb

ln xbe lnb xe

Using the chain rule we get ln lnb xy e b

lnxb b

lnxy b b

sin

Find the equation of the tangent and normal lines to the

curve 3 when .6

xy x

Example

ln lnxbe b

lnu ud dub b bdx dx

cos3Given sin 1 2 , find .w dzz wdw

Example

lnu ud dub b bdx dx

u ud due edx dx

Hence, the generalized derivatives for exponential functions:

lny xye x

yd de xdx dx

1y dyedx

1y

dydx e

1So, lnd xdx x

1lnd duudx u dx

Now we want to find the derivative of logarithmic functions.

The generalized derivative is:

But, ye x

Example

Example

To find the derivative of any logarithmic function of any base, you can use the change of base rule for logs:

logbd xdx

lnln

d xdx b

1 ln

lnd x

b dx

1 1lnb x

Change of Base to e

The generalized derivative formula of the log of any base is:

1 1loglnb

d duudx b u dx

To expand the domain of logarithmic functions we take the absolutevalue of the argument. So lets examine

log , >1b

y x b

It can shown that,

1 1loglnb

d duudx b u dx

The derivative is the same!!

341) log 1 , find .y x y

2

3

sin 12) ln , find .

2 1

xx x e dyydxx

Example

Example

sin

2

1) If , find .x

x

dyy xdx

Example

2)

Example

u ud due edx dx

lnu ud dub b bdx dx

1 1loglnb

d duudx b u dx

1lnd duu

dx u dx

So in conclusion, the generalized derivatives of exponential andlogarithmic functions are: