derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık crot e x start...
TRANSCRIPT
![Page 1: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/1.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Start
Full Screen
Close
Home
Mendel University Brno
Derivatives – the chain rule
Robert Marık
You will differentiate composite functions.
• Full screen button or CTRL+L switshes between window and FullScreen mode.
• Start button gives you a random problem.
• Hint button shows you a hint.
• Solution button shows you a solution.
• Next question button shows another random problem.
• Home button moves here.
c© 2006 [email protected] Revision Date: September 28, 2006 Version 1.0
![Page 2: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/2.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Question
Find y′ for y = (3x − 1)6.
![Page 3: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/3.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Question
Find y′ for y = e−x.
![Page 4: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/4.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Question
Find y′ for y = e1−x2
.
![Page 5: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/5.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Question
Find y′ for y = e4x−1.
![Page 6: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/6.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Question
Find y′ for y = ln(1 − x).
![Page 7: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/7.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Question
Find y′ for y = (x + 3)−3.
![Page 8: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/8.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Question
Find y′ for y = ln(sin x).
![Page 9: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/9.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Question
Find y′ for y = sin(2x).
![Page 10: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/10.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Question
Find y′ for y = sin2 x.
![Page 11: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/11.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Question
Find y′ for y = cos3(2x).
![Page 12: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/12.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Question
Find y′ for y = sin(ex).
![Page 13: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/13.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Question
Find y′ for y = sin(ln x).
![Page 14: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/14.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Question
Find y′ for y = arctg(x2).
![Page 15: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/15.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Question
Find y′ for y = arctg1
x.
![Page 16: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/16.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Question
Find y′ for y = ln(x2 − 1).
![Page 17: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/17.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Question
Find y′ for y = arcsin√
x.
![Page 18: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/18.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Question
Find y′ for y = arctg√
x.
![Page 19: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/19.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Question
Find y′ for y = arctg x2.
![Page 20: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/20.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Question
Find y′ for y = arctg2 x.
![Page 21: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/21.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Question
Find y′ for y = tg 3x.
![Page 22: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/22.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Question
Find y′ for y = tg ln x.
![Page 23: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/23.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Question
Find y′ for y = tg3 x.
![Page 24: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/24.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Answer
y′ =((3x − 1)6)′ = 18(3x − 1)5
![Page 25: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/25.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Answer
y′ =(e−x
)′= −e−x
![Page 26: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/26.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Answer
y′ =(e1−x2
)′= −2xe1−x2
![Page 27: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/27.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Answer
y′ =(e4x−1)′ = 4e4x−1
![Page 28: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/28.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Answer
y′ = (ln(1 − x))′ = − 1
1 − x
![Page 29: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/29.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Answer
y′ =((x + 3)−3)′ = −3(x + 3)−4
![Page 30: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/30.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Answer
y′ = (ln(sin x))′ =cos x
sin x
![Page 31: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/31.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Answer
y′ = (sin(2x))′ = 2 cos(2x)
![Page 32: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/32.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Answer
y′ =(sin2 x
)′= 2 sin x cos x
![Page 33: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/33.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Answer
y′ =(cos3(2x)
)′= −6 cos2(2x) sin(2x)
![Page 34: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/34.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Answer
y′ = (sin(ex))′ = ex cos(ex)
![Page 35: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/35.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Answer
y′ = (sin(ln x))′ =cos(ln x)
x
![Page 36: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/36.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Answer
y′ =(arctg(x2)
)′=
2x
1 + x4
![Page 37: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/37.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Answer
y′ =
(arctg
1
x
)′=
1
1 +( 1
x
)2 ·(− 1
x2
)
![Page 38: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/38.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Answer
y′ =(ln(x2 − 1)
)′=
2x
x2 − 1
![Page 39: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/39.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Answer
y′ =(arcsin
√x)′
=1√
1 − x
1
2
1√x
![Page 40: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/40.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Answer
y′ =(arctg
√x)′
=1
1 + x
1
2
1√x
![Page 41: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/41.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Answer
y′ =(arctg x2)′ =
1
1 + x42x
![Page 42: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/42.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Answer
y′ =(arctg2 x
)′= 2
arctg x
1 + x2
![Page 43: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/43.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Answer
y′ = (tg 3x)′ =3
cos2 3x
![Page 44: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/44.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Answer
y′ = (tg ln x)′ =1
x cos2 ln x
![Page 45: Derivatives -- the chain ruleuser.mendelu.cz/marik/cards/chain-rule.pdfivatives ık croT E X Start Full Screen Close Home Mendel University Brno Derivatives – the chain rule Robert](https://reader033.vdocuments.us/reader033/viewer/2022052811/6084dc43e0fd1457eb45624c/html5/thumbnails/45.jpg)
Derivatives
Robert
Marık
createdusin
gA
croTE X
Hint
Solution
Next question
Home
Answer
y′ =(tg3 x
)′= 3 tg2 x
1
cos2 x