Download - 11X1 T09 03 rules for differentiation (2011)
![Page 1: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/1.jpg)
Rules For Differentiation
![Page 2: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/2.jpg)
Rules For Differentiation(1) y c
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Rules For Differentiation(1) y c
f x c
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Rules For Differentiation(1) y c
f x c
f x h c
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Rules For Differentiation(1) y c
f x c
f x h c
0
limh
c cf xh
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Rules For Differentiation(1) y c
f x c
f x h c
0
limh
c cf xh
0
0limh h
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Rules For Differentiation(1) y c
f x c
f x h c
0
limh
c cf xh
0
0limh h
0lim0
0h
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Rules For Differentiation(1) y c
f x c
f x h c
0
limh
c cf xh
0
0limh h
0lim0
0h
(2) y kx
![Page 9: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/9.jpg)
Rules For Differentiation(1) y c
f x c
f x h c
0
limh
c cf xh
0
0limh h
0lim0
0h
(2) y kx
f x kx
![Page 10: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/10.jpg)
Rules For Differentiation(1) y c
f x c
f x h c
0
limh
c cf xh
0
0limh h
0lim0
0h
(2) y kx
f x kx
f x h k x hkx kh
![Page 11: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/11.jpg)
Rules For Differentiation(1) y c
f x c
f x h c
0
limh
c cf xh
0
0limh h
0lim0
0h
(2) y kx
f x kx
f x h k x hkx kh
0
limh
kx kh kxf xh
![Page 12: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/12.jpg)
Rules For Differentiation(1) y c
f x c
f x h c
0
limh
c cf xh
0
0limh h
0lim0
0h
(2) y kx
f x kx
f x h k x hkx kh
0
limh
kx kh kxf xh
0limh
khh
![Page 13: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/13.jpg)
Rules For Differentiation(1) y c
f x c
f x h c
0
limh
c cf xh
0
0limh h
0lim0
0h
(2) y kx
f x kx
f x h k x hkx kh
0
limh
kx kh kxf xh
0limh
khh
0limh
k
k
![Page 14: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/14.jpg)
(3) ny x
![Page 15: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/15.jpg)
(3) ny x
2 2a b a b a b
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(3) ny x
2 2a b a b a b
3 3 2 2a b a b a ab b
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(3) ny x
2 2a b a b a b
3 3 2 2a b a b a ab b
4 4 3 2 2 3a b a b a a b ab b
![Page 18: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/18.jpg)
(3) ny x
2 2a b a b a b
3 3 2 2a b a b a ab b
4 4 3 2 2 3a b a b a a b ab b
1 2 3 2 2 3 2 1n n n n n n n na b a b a a b a b a b ab b
![Page 19: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/19.jpg)
(3) ny x
2 2a b a b a b
3 3 2 2a b a b a ab b
4 4 3 2 2 3a b a b a a b ab b
1 2 3 2 2 3 2 1n n n n n n n na b a b a a b a b a b ab b
nf x x
![Page 20: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/20.jpg)
(3) ny x
2 2a b a b a b
3 3 2 2a b a b a ab b
4 4 3 2 2 3a b a b a a b ab b
1 2 3 2 2 3 2 1n n n n n n n na b a b a a b a b a b ab b
nf x x nf x h x h
![Page 21: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/21.jpg)
(3) ny x
2 2a b a b a b
3 3 2 2a b a b a ab b
4 4 3 2 2 3a b a b a a b ab b
1 2 3 2 2 3 2 1n n n n n n n na b a b a a b a b a b ab b
nf x x nf x h x h
0
limn n
h
x h xf x
h
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(3) ny x
2 2a b a b a b
3 3 2 2a b a b a ab b
4 4 3 2 2 3a b a b a a b ab b
1 2 3 2 2 3 2 1n n n n n n n na b a b a a b a b a b ab b
nf x x nf x h x h
0
limn n
h
x h xf x
h
1 2 2 1
0lim
n n n n
h
x h x x h x h x x h x x
h
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(3) ny x
2 2a b a b a b
3 3 2 2a b a b a ab b
4 4 3 2 2 3a b a b a a b ab b
1 2 3 2 2 3 2 1n n n n n n n na b a b a a b a b a b ab b
nf x x nf x h x h
0
limn n
h
x h xf x
h
1 2 2 1
0lim
n n n n
h
x h x x h x h x x h x x
h
1 2 2 1
0lim
n n n n
h
h x h x h x x h x x
h
![Page 24: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/24.jpg)
(3) ny x
2 2a b a b a b
3 3 2 2a b a b a ab b
4 4 3 2 2 3a b a b a a b ab b
1 2 3 2 2 3 2 1n n n n n n n na b a b a a b a b a b ab b
nf x x nf x h x h
0
limn n
h
x h xf x
h
1 2 2 1
0lim
n n n n
h
x h x x h x h x x h x x
h
1 2 2 1
0lim
n n n n
h
h x h x h x x h x x
h
1 2 2 1
0lim n n n n
hx h x h x x h x x
![Page 25: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/25.jpg)
1 2 2 1
0lim n n n n
hf x x h x h x x h x x
![Page 26: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/26.jpg)
1 2 2 1
0lim n n n n
hf x x h x h x x h x x
1 1 1 1n n n nx x x x
![Page 27: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/27.jpg)
1 2 2 1
0lim n n n n
hf x x h x h x x h x x
1 1 1 1n n n nx x x x 1nnx
![Page 28: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/28.jpg)
1 2 2 1
0lim n n n n
hf x x h x h x x h x x
1 1 1 1n n n nx x x x 1nnx
1(4) yx
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1 2 2 1
0lim n n n n
hf x x h x h x x h x x
1 1 1 1n n n nx x x x 1nnx
1(4) yx
1f xx
![Page 30: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/30.jpg)
1 2 2 1
0lim n n n n
hf x x h x h x x h x x
1 1 1 1n n n nx x x x 1nnx
1(4) yx
1f xx
1f x hx h
![Page 31: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/31.jpg)
1 2 2 1
0lim n n n n
hf x x h x h x x h x x
1 1 1 1n n n nx x x x 1nnx
1(4) yx
1f xx
1f x hx h
0
1 1
limh
x h xf xh
![Page 32: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/32.jpg)
1 2 2 1
0lim n n n n
hf x x h x h x x h x x
1 1 1 1n n n nx x x x 1nnx
1(4) yx
1f xx
1f x hx h
0
1 1
limh
x h xf xh
0limh
x x hhx x h
![Page 33: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/33.jpg)
1 2 2 1
0lim n n n n
hf x x h x h x x h x x
1 1 1 1n n n nx x x x 1nnx
1(4) yx
1f xx
1f x hx h
0
1 1
limh
x h xf xh
0limh
x x hhx x h
0limh
hhx x h
![Page 34: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/34.jpg)
1 2 2 1
0lim n n n n
hf x x h x h x x h x x
1 1 1 1n n n nx x x x 1nnx
1(4) yx
1f xx
1f x hx h
0
1 1
limh
x h xf xh
0limh
x x hhx x h
0limh
hhx x h
0
1limh x x h
![Page 35: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/35.jpg)
1 2 2 1
0lim n n n n
hf x x h x h x x h x x
1 1 1 1n n n nx x x x 1nnx
1(4) yx
1f xx
1f x hx h
0
1 1
limh
x h xf xh
0limh
x x hhx x h
0limh
hhx x h
0
1limh x x h
21
x
![Page 36: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/36.jpg)
1 2 2 1
0lim n n n n
hf x x h x h x x h x x
1 1 1 1n n n nx x x x 1nnx
1(4) yx
1f xx
1f x hx h
0
1 1
limh
x h xf xh
0limh
x x hhx x h
0limh
hhx x h
0
1limh x x h
21
x
Note:
![Page 37: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/37.jpg)
1 2 2 1
0lim n n n n
hf x x h x h x x h x x
1 1 1 1n n n nx x x x 1nnx
1(4) yx
1f xx
1f x hx h
0
1 1
limh
x h xf xh
0limh
x x hhx x h
0limh
hhx x h
0
1limh x x h
21
x
Note:
1f x x
![Page 38: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/38.jpg)
1 2 2 1
0lim n n n n
hf x x h x h x x h x x
1 1 1 1n n n nx x x x 1nnx
1(4) yx
1f xx
1f x hx h
0
1 1
limh
x h xf xh
0limh
x x hhx x h
0limh
hhx x h
0
1limh x x h
21
x
Note:
1f x x
2f x x
![Page 39: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/39.jpg)
1 2 2 1
0lim n n n n
hf x x h x h x x h x x
1 1 1 1n n n nx x x x 1nnx
1(4) yx
1f xx
1f x hx h
0
1 1
limh
x h xf xh
0limh
x x hhx x h
0limh
hhx x h
0
1limh x x h
21
x
Note:
1f x x
2f x x
21
x
![Page 40: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/40.jpg)
(5) y x
![Page 41: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/41.jpg)
(5) y x
f x x
![Page 42: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/42.jpg)
(5) y x
f x x
f x h x h
![Page 43: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/43.jpg)
(5) y x
f x x
f x h x h 0
limh
x h xf xh
![Page 44: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/44.jpg)
(5) y x
f x x
f x h x h 0
limh
x h xf xh
x h xx h x
![Page 45: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/45.jpg)
(5) y x
f x x
f x h x h 0
limh
x h xf xh
x h xx h x
0limh
x h xh x h x
![Page 46: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/46.jpg)
(5) y x
f x x
f x h x h 0
limh
x h xf xh
x h xx h x
0limh
x h xh x h x
0limh
hh x h x
![Page 47: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/47.jpg)
(5) y x
f x x
f x h x h 0
limh
x h xf xh
x h xx h x
0limh
x h xh x h x
0limh
hh x h x
0
1limh x h x
![Page 48: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/48.jpg)
(5) y x
f x x
f x h x h 0
limh
x h xf xh
x h xx h x
0limh
x h xh x h x
0limh
hh x h x
0
1limh x h x
1x x
![Page 49: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/49.jpg)
(5) y x
f x x
f x h x h 0
limh
x h xf xh
x h xx h x
0limh
x h xh x h x
0limh
hh x h x
0
1limh x h x
1x x
12 x
![Page 50: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/50.jpg)
(5) y x
f x x
f x h x h 0
limh
x h xf xh
x h xx h x
0limh
x h xh x h x
0limh
hh x h x
0
1limh x h x
1x x
12 x
Note:
![Page 51: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/51.jpg)
(5) y x
f x x
f x h x h 0
limh
x h xf xh
x h xx h x
0limh
x h xh x h x
0limh
hh x h x
0
1limh x h x
1x x
12 x
Note:
12f x x
![Page 52: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/52.jpg)
(5) y x
f x x
f x h x h 0
limh
x h xf xh
x h xx h x
0limh
x h xh x h x
0limh
hh x h x
0
1limh x h x
1x x
12 x
Note:
12f x x
121
2f x x
![Page 53: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/53.jpg)
(5) y x
f x x
f x h x h 0
limh
x h xf xh
x h xx h x
0limh
x h xh x h x
0limh
hh x h x
0
1limh x h x
1x x
12 x
Note:
12f x x
121
2f x x
12 x
![Page 54: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/54.jpg)
e.g. 7i y
![Page 55: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/55.jpg)
e.g. 7i y
0dydx
![Page 56: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/56.jpg)
e.g. 7i y
0dydx
37ii y x
![Page 57: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/57.jpg)
e.g. 7i y
0dydx
37ii y x
37dydx
![Page 58: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/58.jpg)
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x
![Page 59: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/59.jpg)
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
![Page 60: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/60.jpg)
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
23 6 2iv y x x
![Page 61: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/61.jpg)
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
23 6 2iv y x x
6 6dy xdx
![Page 62: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/62.jpg)
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
23 6 2iv y x x
6 6dy xdx
22 1v y x
![Page 63: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/63.jpg)
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
23 6 2iv y x x
6 6dy xdx
22 1v y x 24 4 1x x
![Page 64: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/64.jpg)
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
23 6 2iv y x x
6 6dy xdx
22 1v y x 24 4 1x x
8 4dy xdx
![Page 65: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/65.jpg)
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
23 6 2iv y x x
6 6dy xdx
22 1v y x 24 4 1x x
8 4dy xdx
21 3vi y xx
![Page 66: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/66.jpg)
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
23 6 2iv y x x
6 6dy xdx
22 1v y x 24 4 1x x
8 4dy xdx
21 3vi y xx
23x x
![Page 67: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/67.jpg)
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
23 6 2iv y x x
6 6dy xdx
22 1v y x 24 4 1x x
8 4dy xdx
21 3vi y xx
23x x 33 2dy x
dx
![Page 68: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/68.jpg)
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
23 6 2iv y x x
6 6dy xdx
22 1v y x 24 4 1x x
8 4dy xdx
21 3vi y xx
23x x 33 2dy x
dx
323x
![Page 69: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/69.jpg)
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
23 6 2iv y x x
6 6dy xdx
22 1v y x 24 4 1x x
8 4dy xdx
21 3vi y xx
23x x 33 2dy x
dx
323x
2 vii y x x
![Page 70: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/70.jpg)
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
23 6 2iv y x x
6 6dy xdx
22 1v y x 24 4 1x x
8 4dy xdx
21 3vi y xx
23x x 33 2dy x
dx
323x
2 vii y x x52x
![Page 71: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/71.jpg)
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
23 6 2iv y x x
6 6dy xdx
22 1v y x 24 4 1x x
8 4dy xdx
21 3vi y xx
23x x 33 2dy x
dx
323x
2 vii y x x52x
325
2dy xdx
![Page 72: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/72.jpg)
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
23 6 2iv y x x
6 6dy xdx
22 1v y x 24 4 1x x
8 4dy xdx
21 3vi y xx
23x x 33 2dy x
dx
323x
2 vii y x x52x
325
2dy xdx
52
x x
![Page 73: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/73.jpg)
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
23 6 2iv y x x
6 6dy xdx
22 1v y x 24 4 1x x
8 4dy xdx
21 3vi y xx
23x x 33 2dy x
dx
323x
2 vii y x x52x
325
2dy xdx
52
x x
3If 3, find 2viii f x x
f
![Page 74: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/74.jpg)
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
23 6 2iv y x x
6 6dy xdx
22 1v y x 24 4 1x x
8 4dy xdx
21 3vi y xx
23x x 33 2dy x
dx
323x
2 vii y x x52x
325
2dy xdx
52
x x
3If 3, find 2viii f x x
f
3 3f x x
![Page 75: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/75.jpg)
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
23 6 2iv y x x
6 6dy xdx
22 1v y x 24 4 1x x
8 4dy xdx
21 3vi y xx
23x x 33 2dy x
dx
323x
2 vii y x x52x
325
2dy xdx
52
x x
3If 3, find 2viii f x x
f
3 3f x x
23f x x
![Page 76: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/76.jpg)
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
23 6 2iv y x x
6 6dy xdx
22 1v y x 24 4 1x x
8 4dy xdx
21 3vi y xx
23x x 33 2dy x
dx
323x
2 vii y x x52x
325
2dy xdx
52
x x
3If 3, find 2viii f x x
f
3 3f x x
23f x x
22 3 2f
![Page 77: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/77.jpg)
e.g. 7i y
0dydx
37ii y x
37dydx
10 iii y x910dy x
dx
23 6 2iv y x x
6 6dy xdx
22 1v y x 24 4 1x x
8 4dy xdx
21 3vi y xx
23x x 33 2dy x
dx
323x
2 vii y x x52x
325
2dy xdx
52
x x
3If 3, find 2viii f x x
f
3 3f x x
23f x x
22 3 2f
12
![Page 78: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/78.jpg)
3 2 Find the equation of the tangent to the curve 5 6 2 at the point 1,1xix y x x
![Page 79: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/79.jpg)
3 2 Find the equation of the tangent to the curve 5 6 2 at the point 1,1xix y x x
3 25 6 2y x x
![Page 80: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/80.jpg)
3 2 Find the equation of the tangent to the curve 5 6 2 at the point 1,1xix y x x
3 25 6 2y x x 215 12dy x x
dx
![Page 81: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/81.jpg)
3 2 Find the equation of the tangent to the curve 5 6 2 at the point 1,1xix y x x
3 25 6 2y x x 215 12dy x x
dx
2when 1, 15 1 12 1dyxdx
![Page 82: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/82.jpg)
3 2 Find the equation of the tangent to the curve 5 6 2 at the point 1,1xix y x x
3 25 6 2y x x 215 12dy x x
dx
2when 1, 15 1 12 1dyxdx
3
![Page 83: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/83.jpg)
3 2 Find the equation of the tangent to the curve 5 6 2 at the point 1,1xix y x x
3 25 6 2y x x 215 12dy x x
dx
2when 1, 15 1 12 1dyxdx
3required slope 3
![Page 84: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/84.jpg)
3 2 Find the equation of the tangent to the curve 5 6 2 at the point 1,1xix y x x
3 25 6 2y x x 215 12dy x x
dx
2when 1, 15 1 12 1dyxdx
3required slope 3
1 3 1y x
![Page 85: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/85.jpg)
3 2 Find the equation of the tangent to the curve 5 6 2 at the point 1,1xix y x x
3 25 6 2y x x 215 12dy x x
dx
2when 1, 15 1 12 1dyxdx
3required slope 3
1 3 1y x
1 3 3y x
![Page 86: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/86.jpg)
3 2 Find the equation of the tangent to the curve 5 6 2 at the point 1,1xix y x x
3 25 6 2y x x 215 12dy x x
dx
2when 1, 15 1 12 1dyxdx
3required slope 3
1 3 1y x
1 3 3y x
3 2 0x y
![Page 87: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/87.jpg)
3 2 Find the equation of the tangent to the curve 5 6 2 at the point 1,1xix y x x
3 25 6 2y x x 215 12dy x x
dx
2when 1, 15 1 12 1dyxdx
3required slope 3
1 3 1y x
1 3 3y x
3 2 0x y
3 Find the points on the curve 12 where the tangents are horizontalx y x x
![Page 88: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/88.jpg)
3 2 Find the equation of the tangent to the curve 5 6 2 at the point 1,1xix y x x
3 25 6 2y x x 215 12dy x x
dx
2when 1, 15 1 12 1dyxdx
3required slope 3
1 3 1y x
1 3 3y x
3 2 0x y
3 Find the points on the curve 12 where the tangents are horizontalx y x x
3 12y x x
![Page 89: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/89.jpg)
3 2 Find the equation of the tangent to the curve 5 6 2 at the point 1,1xix y x x
3 25 6 2y x x 215 12dy x x
dx
2when 1, 15 1 12 1dyxdx
3required slope 3
1 3 1y x
1 3 3y x
3 2 0x y
3 Find the points on the curve 12 where the tangents are horizontalx y x x
3 12y x x
23 12dy xdx
![Page 90: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/90.jpg)
3 2 Find the equation of the tangent to the curve 5 6 2 at the point 1,1xix y x x
3 25 6 2y x x 215 12dy x x
dx
2when 1, 15 1 12 1dyxdx
3required slope 3
1 3 1y x
1 3 3y x
3 2 0x y
3 Find the points on the curve 12 where the tangents are horizontalx y x x
3 12y x x
23 12dy xdx
tangents are horizontal when 0dydx
![Page 91: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/91.jpg)
3 2 Find the equation of the tangent to the curve 5 6 2 at the point 1,1xix y x x
3 25 6 2y x x 215 12dy x x
dx
2when 1, 15 1 12 1dyxdx
3required slope 3
1 3 1y x
1 3 3y x
3 2 0x y
3 Find the points on the curve 12 where the tangents are horizontalx y x x
3 12y x x
23 12dy xdx
tangents are horizontal when 0dydx
2i.e. 3 12 0x
![Page 92: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/92.jpg)
3 2 Find the equation of the tangent to the curve 5 6 2 at the point 1,1xix y x x
3 25 6 2y x x 215 12dy x x
dx
2when 1, 15 1 12 1dyxdx
3required slope 3
1 3 1y x
1 3 3y x
3 2 0x y
3 Find the points on the curve 12 where the tangents are horizontalx y x x
3 12y x x
23 12dy xdx
tangents are horizontal when 0dydx
2i.e. 3 12 0x
2 42
xx
![Page 93: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/93.jpg)
3 2 Find the equation of the tangent to the curve 5 6 2 at the point 1,1xix y x x
3 25 6 2y x x 215 12dy x x
dx
2when 1, 15 1 12 1dyxdx
3required slope 3
1 3 1y x
1 3 3y x
3 2 0x y
3 Find the points on the curve 12 where the tangents are horizontalx y x x
3 12y x x
23 12dy xdx
tangents are horizontal when 0dydx
2i.e. 3 12 0x
2 42
xx
tangents are horizontal at 2,16 and 2, 16
![Page 94: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/94.jpg)
A normal is a line perpendicular to the tangent at the point of contact
![Page 95: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/95.jpg)
A normal is a line perpendicular to the tangent at the point of contacty
x
y f x
![Page 96: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/96.jpg)
A normal is a line perpendicular to the tangent at the point of contacty
x
y f x
tangent
![Page 97: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/97.jpg)
A normal is a line perpendicular to the tangent at the point of contacty
x
y f x
tangentnormal
![Page 98: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/98.jpg)
A normal is a line perpendicular to the tangent at the point of contacty
x
y f x
tangentnormal
2 Find the equation of the normal to the curve 4 3 2 at the point 3,29xi y x x
![Page 99: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/99.jpg)
A normal is a line perpendicular to the tangent at the point of contacty
x
y f x
tangentnormal
2 Find the equation of the normal to the curve 4 3 2 at the point 3,29xi y x x
24 3 2y x x
![Page 100: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/100.jpg)
A normal is a line perpendicular to the tangent at the point of contacty
x
y f x
tangentnormal
2 Find the equation of the normal to the curve 4 3 2 at the point 3,29xi y x x
24 3 2y x x
8 3dy xdx
![Page 101: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/101.jpg)
A normal is a line perpendicular to the tangent at the point of contacty
x
y f x
tangentnormal
2 Find the equation of the normal to the curve 4 3 2 at the point 3,29xi y x x
24 3 2y x x
8 3dy xdx
when 3, 8 3 3dyxdx
![Page 102: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/102.jpg)
A normal is a line perpendicular to the tangent at the point of contacty
x
y f x
tangentnormal
2 Find the equation of the normal to the curve 4 3 2 at the point 3,29xi y x x
24 3 2y x x
8 3dy xdx
when 3, 8 3 3dyxdx
21
![Page 103: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/103.jpg)
A normal is a line perpendicular to the tangent at the point of contacty
x
y f x
tangentnormal
2 Find the equation of the normal to the curve 4 3 2 at the point 3,29xi y x x
24 3 2y x x
8 3dy xdx
when 3, 8 3 3dyxdx
21 1 required slope21
![Page 104: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/104.jpg)
A normal is a line perpendicular to the tangent at the point of contacty
x
y f x
tangentnormal
2 Find the equation of the normal to the curve 4 3 2 at the point 3,29xi y x x
24 3 2y x x
8 3dy xdx
when 3, 8 3 3dyxdx
21 1 required slope21
129 321
y x
![Page 105: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/105.jpg)
A normal is a line perpendicular to the tangent at the point of contacty
x
y f x
tangentnormal
2 Find the equation of the normal to the curve 4 3 2 at the point 3,29xi y x x
24 3 2y x x
8 3dy xdx
when 3, 8 3 3dyxdx
21 1 required slope21
129 321
y x
21 609 3y x
![Page 106: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/106.jpg)
A normal is a line perpendicular to the tangent at the point of contacty
x
y f x
tangentnormal
2 Find the equation of the normal to the curve 4 3 2 at the point 3,29xi y x x
24 3 2y x x
8 3dy xdx
when 3, 8 3 3dyxdx
21 1 required slope21
129 321
y x
21 609 3y x
21 612 0x y
![Page 107: 11X1 T09 03 rules for differentiation (2011)](https://reader034.vdocument.in/reader034/viewer/2022052311/55895ac9d8b42a81178b4633/html5/thumbnails/107.jpg)
Exercise 7C; 1ace etc, 2ace etc, 3ace etc, 4bd, 5bdfh, 8bd, 9bd, 10ac, 12, 13b, 16, 21
Exercise 7D; 2ac, 3bd, 4ace, 6c, 7b, 11a, 13aei, 18, 22