3.9: Derivatives of Exponential and Logarithmic Functions Mt. Rushmore, South Dakota

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2007

Look at the graph of xy e

The slope at x=0 appears to be 1.

If we assume this to be true, then:

0 0

0lim 1

h

h

e e

h

definition of derivative

Now we attempt to find a general formula for the derivative of using the definition.

xy e

0

limx h x

x

h

d e ee

dx h

0lim

x h x

h

e e e

h

0

1lim

hx

h

ee

h

0

1lim

hx

h

ee

h

1xe xe

This is the slope at x=0, which we have assumed to be 1.

x xde e

dx

xe is its own derivative!

If we incorporate the chain rule:

u ud due e

dx dx

We can now use this formula to find the derivative ofxa

xda

dx

ln xade

dx( and are inverse functions.)

xe ln x

lnx ade

dx

ln lnx a de x a

dx (chain rule)

( is a constant.)ln a

xda

dx

ln xade

dx

lnx ade

dx

ln lnx a de x a

dx

ln lnx ae a

lnxa a

Incorporating the chain rule:

lnu ud dua a a

dx dx

So far today we have:

u ud due e

dx dx lnu ud du

a a adx dx

Now it is relatively easy to find the derivative of .ln x

lny x

ye x

yd de x

dx dx

1y dyedx

1y

dy

dx e

1ln

dx

dx x

1ln

d duu

dx u dx

To find the derivative of a common log function, you could just use the change of base rule for logs:

logd

xdx

ln

ln10

d x

dx

1ln

ln10

dx

dx

1 1

ln10 x

The formula for the derivative of a log of any base other than e is:

1log

lna

d duu

dx u a dx

u ud due e

dx dx lnu ud du

a a adx dx

1log

lna

d duu

dx u a dx

1ln

d duu

dx u dx