Derivatives of Exponential FunctionsLesson 4.4

An Interesting Function

Consider the function y = ax • Let a = 2 • Graph the function and it's derivative

2

Try the same thing with

a = 3a = 2.5a = 2.7

Try the same thing with

a = 3a = 2.5a = 2.7

An Interesting Function

Consider that there might be a function that is its own derivative

Try f (x) = ex

Conclusion:

3

x xxD e e

Derivative of ax

When f(x) = ax

Consider using the definition of derivative

4

0

0

0

lim

lim

1lim

x x h x

h

x h x

h

hx

h

d a a a

dx h

a a a

h

aa

h

What is the justification for

each step?

What is the justification for

each step?

Derivative of ax

Now to deal with the right hand side of the expression

Try graphing

• Look familiar?

5

0

1lim

hx

h

aa

h

.0001 1

.0001

xy

0

1lim ln

h

h

aa

h

Derivative of ax

Conclusion

When y = ag(x)

• Use chain rule

Similarly for y = eg(x)

6

ln( )x xxD a a a

( )ln '( )g x

dy dy du

dx du dx

a a g x

( ) ( ) '( )g x g xxD e e g x

Practice

Try taking the derivatives of the following exponential functions

7

2xy e5( ) 8 xf x

22 2xy x e

0.5

500( )

12 5 xf x

e

Assignments

Lesson 4.4

Page 279

Exercises 1 – 61 EOO

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