Differentiation Higher Mathematics Next.
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![Page 1: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/1.jpg)
Differentiation
Higher Mathematics
www.maths4scotland.co.uk
Next
![Page 2: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/2.jpg)
Calculus Revision
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Differentiate24 3 7x x
8 3x
![Page 3: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/3.jpg)
Calculus Revision
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Differentiate
3 26 3 9x x x
x
3 26 3 9x x x
x x x x
2 16 3 9x x x
Split up
Straight line form
22 6 9x x Differentiate
![Page 4: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/4.jpg)
Calculus Revision
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Differentiate
3 1
2 22 5x x
1 3
2 23 12 2
2 5x x
1 3
2 25
23x x
![Page 5: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/5.jpg)
Calculus Revision
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Differentiate2
3 1xx
1 1
2 23 2 1x x
Straight line form
Differentiate
1 3
2 21
2
32
2x x
1 3
2 23
2x x
![Page 6: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/6.jpg)
Calculus Revision
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Differentiate (3 5)( 2)x x
23 6 5 10x x x Multiply out
Differentiate 6 1x
23 10x x
![Page 7: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/7.jpg)
Calculus Revision
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Differentiate 2 8003 r
r
2 13 800r r Straight line form
Differentiate26 ( 1)800r r
![Page 8: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/8.jpg)
Calculus Revision
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Differentiate2( 2 )x x x
multiply out3 22x x
differentiate23 4x x
![Page 9: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/9.jpg)
Calculus Revision
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Differentiate3
1 1
x x
1 3x x Straight line form
Differentiate2 4( 3)x x
2 43x x
![Page 10: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/10.jpg)
Calculus Revision
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Differentiate 2x x x
1
22x x xStraight line form
multiply out
Differentiate
5 3
2 2x x3 1
2 25 3
2 2x x
![Page 11: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/11.jpg)
Calculus Revision
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Differentiate 2 4x x
multiply out
Simplify
4 8 2x x x
Differentiate
6 8x x
Straight line form
1
26 8x x 1
23 1x
![Page 12: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/12.jpg)
Calculus Revision
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Differentiate5( 2)x
Chain rule45( 2) 1x
45( 2)x Simplify
![Page 13: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/13.jpg)
Calculus Revision
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Differentiate3(5 1)x
Chain Rule23(5 1) 5x
215(5 1)x Simplify
![Page 14: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/14.jpg)
Calculus Revision
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Differentiate2 4(5 3 2)x x
Chain Rule 2 34(5 3 2) 10 3x x x
![Page 15: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/15.jpg)
Calculus Revision
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Differentiate
5
2(7 1)x
Chain Rule
Simplify
3
25
2(7 1) 7x
3
235
2(7 1)x
![Page 16: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/16.jpg)
Calculus Revision
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Differentiate
1
2(2 5)x
Chain Rule
Simplify
3
21
2(2 5) 2x
3
2(2 5)x
![Page 17: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/17.jpg)
Calculus Revision
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Differentiate 3 1x
Chain Rule
Simplify
Straight line form 1
23 1x
1
21
23 1 3x
1
23
23 1x
![Page 18: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/18.jpg)
Calculus Revision
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Differentiate 42
5
3x
Chain Rule
Simplify
Straight line form2 45( 3)x
2 520( 3) 2x x
2 540 ( 3)x x
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Calculus Revision
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Differentiate1
2 1x
Chain Rule
Simplify
Straight line form
1
2(2 1)x
3
21
2(2 1) 2x
3
2(2 1)x
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Calculus Revision
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Differentiate2
2cos sin3
x x
22sin cos
3x x
![Page 21: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/21.jpg)
Calculus Revision
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Differentiate 3cos x
3sin x
![Page 22: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/22.jpg)
Calculus Revision
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Differentiate cos 4 2sin 2x x
4sin 4 4cos 2x x
![Page 23: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/23.jpg)
Calculus Revision
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Differentiate4cos x
34cos sinx x
![Page 24: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/24.jpg)
Calculus Revision
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Differentiate sin x
Chain Rule
Simplify
Straight line form
1
2(sin )x1
21
2(sin ) cosx x
1
21
2cos (sin )x x
1
2
cos
sin
x
x
![Page 25: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/25.jpg)
Calculus Revision
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Differentiate sin(3 2 )x
Chain Rule
Simplify
cos(3 2 ) ( 2)x
2cos(3 2 )x
![Page 26: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/26.jpg)
Calculus Revision
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Differentiatecos5
2
x
Chain Rule
Simplify
1
2cos5xStraight line form
1
2sin 5 5x
5
2sin 5x
![Page 27: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/27.jpg)
Calculus Revision
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Differentiate ( ) cos(2 ) 3sin(4 )f x x x
( ) 2sin(2 ) 12cos(4 )f x x x
![Page 28: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/28.jpg)
Calculus Revision
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Differentiate2 43200
( )A x xx
Chain Rule
Simplify
Straight line form2 1( ) 43200A x x x
2( ) 2 43200A x x x
2
43200( ) 2A x x
x
![Page 29: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/29.jpg)
Calculus Revision
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Differentiate 2
2( )f x x
x
Differentiate
Straight line form
122( ) 2f x x x
1321
2( ) 4f x x x
![Page 30: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/30.jpg)
Calculus Revision
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Differentiate 3 22 7 4 4y x x x
26 14 4y x x
![Page 31: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/31.jpg)
Calculus Revision
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Differentiate 62siny x
Chain Rule
Simplify
62 cos 1
dyx
dx
62 cos
dyx
dx
![Page 32: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/32.jpg)
Calculus Revision
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Differentiate3
(8 )4
A a a
multiply out
Differentiate
236
4A a a
36
2A a
![Page 33: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/33.jpg)
Calculus Revision
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Differentiate
13 2( ) (8 )f x x
Chain Rule
Simplify
13 221
2( ) (8 ) ( 2 )f x x x
12 3 2( ) (8 )f x x x
![Page 34: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/34.jpg)
Calculus Revision
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Differentiate16
, 0y x xx
Differentiate
Straight line form1
216y x x
3
21 8dy
xdx
![Page 35: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/35.jpg)
Calculus Revision
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Differentiate 23 3 16( )
2A x x
x
Straight line form
Multiply out23 3 3 3 16
( )2 2
A x xx
Differentiate
2 13 3( ) 24 3
2A x x x
2( ) 3 3 24 3A x x x
![Page 36: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/36.jpg)
Calculus Revision
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Differentiate 1
2( ) 5 4f x x
Chain Rule
Simplify
1
21
2( ) 5 4 5f x x
1
25
2( ) 5 4f x x
![Page 37: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/37.jpg)
Calculus Revision
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Differentiate 216( ) 240
3A x x x
32( ) 240
3A x x
![Page 38: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/38.jpg)
Calculus Revision
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Differentiate 2( ) 3 (2 1)f x x x
Multiply out
Differentiate
3 2( ) 6 3f x x x
2( ) 18 6f x x x
![Page 39: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/39.jpg)
Calculus Revision
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Differentiate2 2( ) cos sinf x x x
( ) 2 cos sin 2sin cosf x x x x x Chain Rule
Simplify ( ) 4 cos sinf x x x
![Page 40: Differentiation Higher Mathematics Next.](https://reader035.fdocuments.net/reader035/viewer/2022062216/56649dac5503460f94a9c0b5/html5/thumbnails/40.jpg)
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