Derivatives of Exponential Functions Lesson 4.4. An Interesting Function Consider the function y = a...
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Transcript of Derivatives of Exponential Functions Lesson 4.4. An Interesting Function Consider the function y = a...
![Page 1: Derivatives of Exponential Functions Lesson 4.4. An Interesting Function Consider the function y = a x Let a = 2 Graph the function and it's derivative.](https://reader030.fdocuments.in/reader030/viewer/2022032607/56649ecb5503460f94bd984f/html5/thumbnails/1.jpg)
Derivatives of Exponential FunctionsLesson 4.4
![Page 2: Derivatives of Exponential Functions Lesson 4.4. An Interesting Function Consider the function y = a x Let a = 2 Graph the function and it's derivative.](https://reader030.fdocuments.in/reader030/viewer/2022032607/56649ecb5503460f94bd984f/html5/thumbnails/2.jpg)
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
![Page 3: Derivatives of Exponential Functions Lesson 4.4. An Interesting Function Consider the function y = a x Let a = 2 Graph the function and it's derivative.](https://reader030.fdocuments.in/reader030/viewer/2022032607/56649ecb5503460f94bd984f/html5/thumbnails/3.jpg)
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
![Page 4: Derivatives of Exponential Functions Lesson 4.4. An Interesting Function Consider the function y = a x Let a = 2 Graph the function and it's derivative.](https://reader030.fdocuments.in/reader030/viewer/2022032607/56649ecb5503460f94bd984f/html5/thumbnails/4.jpg)
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?
![Page 5: Derivatives of Exponential Functions Lesson 4.4. An Interesting Function Consider the function y = a x Let a = 2 Graph the function and it's derivative.](https://reader030.fdocuments.in/reader030/viewer/2022032607/56649ecb5503460f94bd984f/html5/thumbnails/5.jpg)
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
![Page 6: Derivatives of Exponential Functions Lesson 4.4. An Interesting Function Consider the function y = a x Let a = 2 Graph the function and it's derivative.](https://reader030.fdocuments.in/reader030/viewer/2022032607/56649ecb5503460f94bd984f/html5/thumbnails/6.jpg)
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
![Page 7: Derivatives of Exponential Functions Lesson 4.4. An Interesting Function Consider the function y = a x Let a = 2 Graph the function and it's derivative.](https://reader030.fdocuments.in/reader030/viewer/2022032607/56649ecb5503460f94bd984f/html5/thumbnails/7.jpg)
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
![Page 8: Derivatives of Exponential Functions Lesson 4.4. An Interesting Function Consider the function y = a x Let a = 2 Graph the function and it's derivative.](https://reader030.fdocuments.in/reader030/viewer/2022032607/56649ecb5503460f94bd984f/html5/thumbnails/8.jpg)
Assignments
Lesson 4.4
Page 279
Exercises 1 – 61 EOO
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