Home » Archimedes archive » Archimedes World » AW-1996-07.adf » !Ignotum_Ignotum » !Ignotum/Formulae/Diff
!Ignotum/Formulae/Diff
This website contains an archive of files for the Acorn Electron, BBC Micro, Acorn Archimedes, Commodore 16 and Commodore 64 computers, which Dominic Ford has rescued from his private collection of floppy disks and cassettes.
Some of these files were originally commercial releases in the 1980s and 1990s, but they are now widely available online. I assume that copyright over them is no longer being asserted. If you own the copyright and would like files to be removed, please contact me.
| Tape/disk: | Home » Archimedes archive » Archimedes World » AW-1996-07.adf » !Ignotum_Ignotum |
| Filename: | !Ignotum/Formulae/Diff |
| Read OK: | ✔ |
| File size: | 0599 bytes |
| Load address: | 0000 |
| Exec address: | 0000 |
File contents
# Maths > Differentiation sinx>cosx '>' means 'goes to' rather than 'greater than'. cosx>-sinx tanx>sec�x secx>secxtanx cosecx>-cosecxcotx cotx>-cosec�x e^x>e^x Where e is the constant 2.718. This does not change when differentiated. However, e^ax>ae^ax. arcsinx>1/(1-x�)^� Arcsine=inverse sine, also written as sin^(-1)x. This is not the same as 1/sinx which would be written as (sinx)^-1. arccosx>-1/(1-x�)^� Arccosine=inverse cosine, also written as cos^(-1)x. This is not the same as 1/cosx which would be written as (cosx)^-1. arctanx>1/(1+x�) Arctan=inverse tan, also written as tan^(-1)x. This is not the same as 1/tanx which would be written as (tanx)^-1. logx>1/x (ax^b+c)^d>d(abx^(b-1))(ax^b+c)^(d-1) To differentiate a bracket such as this with respect to x, you would bring down the power (d), differentiate what is in the bracket and multiply these by the bracket with the power reduced by 1. dy/dx=(dy/du)(du/dx) Where dy/dx is the differential of the equation y=f(x). This is called the chain rule and it is not as obvious as it first seems as they are NOT fractions, but they can sometimes be treated as if they are. uv>v(du/dx)+u(dv/dx) This is called the product rule, where u and v are both functions of x. u/v>(v(du/dx)-u(dv/dx))/v� This is called the quotient rule where u and v are both functions of x.
00000000 23 20 4d 61 74 68 73 20 3e 20 44 69 66 66 65 72 |# Maths > Differ| 00000010 65 6e 74 69 61 74 69 6f 6e 0a 73 69 6e 78 3e 63 |entiation.sinx>c| 00000020 6f 73 78 0a 27 3e 27 20 6d 65 61 6e 73 20 27 67 |osx.'>' means 'g| 00000030 6f 65 73 20 74 6f 27 20 72 61 74 68 65 72 20 74 |oes to' rather t| 00000040 68 61 6e 20 27 67 72 65 61 74 65 72 0a 74 68 61 |han 'greater.tha| 00000050 6e 27 2e 0a 0a 0a 0a 63 6f 73 78 3e 2d 73 69 6e |n'.....cosx>-sin| 00000060 78 0a 0a 0a 0a 0a 0a 74 61 6e 78 3e 73 65 63 b2 |x......tanx>sec.| 00000070 78 0a 0a 0a 0a 0a 0a 73 65 63 78 3e 73 65 63 78 |x......secx>secx| 00000080 74 61 6e 78 0a 0a 0a 0a 0a 0a 63 6f 73 65 63 78 |tanx......cosecx| 00000090 3e 2d 63 6f 73 65 63 78 63 6f 74 78 0a 0a 0a 0a |>-cosecxcotx....| 000000a0 0a 0a 63 6f 74 78 3e 2d 63 6f 73 65 63 b2 78 0a |..cotx>-cosec.x.| 000000b0 0a 0a 0a 0a 0a 65 5e 78 3e 65 5e 78 0a 57 68 65 |.....e^x>e^x.Whe| 000000c0 72 65 20 65 20 69 73 20 74 68 65 20 63 6f 6e 73 |re e is the cons| 000000d0 74 61 6e 74 20 32 2e 37 31 38 2e 20 54 68 69 73 |tant 2.718. This| 000000e0 20 64 6f 65 73 0a 6e 6f 74 20 63 68 61 6e 67 65 | does.not change| 000000f0 20 77 68 65 6e 20 64 69 66 66 65 72 65 6e 74 69 | when differenti| 00000100 61 74 65 64 2e 20 48 6f 77 65 76 65 72 2c 0a 65 |ated. However,.e| 00000110 5e 61 78 3e 61 65 5e 61 78 2e 0a 0a 0a 61 72 63 |^ax>ae^ax....arc| 00000120 73 69 6e 78 3e 31 2f 28 31 2d 78 b2 29 5e bd 0a |sinx>1/(1-x.)^..| 00000130 41 72 63 73 69 6e 65 3d 69 6e 76 65 72 73 65 20 |Arcsine=inverse | 00000140 73 69 6e 65 2c 20 61 6c 73 6f 20 77 72 69 74 74 |sine, also writt| 00000150 65 6e 20 61 73 0a 73 69 6e 5e 28 2d 31 29 78 2e |en as.sin^(-1)x.| 00000160 20 54 68 69 73 20 69 73 20 6e 6f 74 20 74 68 65 | This is not the| 00000170 20 73 61 6d 65 20 61 73 20 31 2f 73 69 6e 78 0a | same as 1/sinx.| 00000180 77 68 69 63 68 20 77 6f 75 6c 64 20 62 65 20 77 |which would be w| 00000190 72 69 74 74 65 6e 20 61 73 20 28 73 69 6e 78 29 |ritten as (sinx)| 000001a0 5e 2d 31 2e 0a 0a 0a 61 72 63 63 6f 73 78 3e 2d |^-1....arccosx>-| 000001b0 31 2f 28 31 2d 78 b2 29 5e bd 0a 41 72 63 63 6f |1/(1-x.)^..Arcco| 000001c0 73 69 6e 65 3d 69 6e 76 65 72 73 65 20 63 6f 73 |sine=inverse cos| 000001d0 69 6e 65 2c 20 61 6c 73 6f 20 77 72 69 74 74 65 |ine, also writte| 000001e0 6e 20 61 73 0a 63 6f 73 5e 28 2d 31 29 78 2e 20 |n as.cos^(-1)x. | 000001f0 54 68 69 73 20 69 73 20 6e 6f 74 20 74 68 65 20 |This is not the | 00000200 73 61 6d 65 20 61 73 20 31 2f 63 6f 73 78 0a 77 |same as 1/cosx.w| 00000210 68 69 63 68 20 77 6f 75 6c 64 20 62 65 20 77 72 |hich would be wr| 00000220 69 74 74 65 6e 20 61 73 20 28 63 6f 73 78 29 5e |itten as (cosx)^| 00000230 2d 31 2e 0a 0a 0a 61 72 63 74 61 6e 78 3e 31 2f |-1....arctanx>1/| 00000240 28 31 2b 78 b2 29 0a 41 72 63 74 61 6e 3d 69 6e |(1+x.).Arctan=in| 00000250 76 65 72 73 65 20 74 61 6e 2c 20 61 6c 73 6f 20 |verse tan, also | 00000260 77 72 69 74 74 65 6e 20 61 73 0a 74 61 6e 5e 28 |written as.tan^(| 00000270 2d 31 29 78 2e 20 54 68 69 73 20 69 73 20 6e 6f |-1)x. This is no| 00000280 74 20 74 68 65 20 73 61 6d 65 20 61 73 20 31 2f |t the same as 1/| 00000290 74 61 6e 78 0a 77 68 69 63 68 20 77 6f 75 6c 64 |tanx.which would| 000002a0 20 62 65 20 77 72 69 74 74 65 6e 20 61 73 20 28 | be written as (| 000002b0 74 61 6e 78 29 5e 2d 31 2e 0a 0a 0a 6c 6f 67 78 |tanx)^-1....logx| 000002c0 3e 31 2f 78 0a 0a 0a 0a 0a 0a 28 61 78 5e 62 2b |>1/x......(ax^b+| 000002d0 63 29 5e 64 3e 64 28 61 62 78 5e 28 62 2d 31 29 |c)^d>d(abx^(b-1)| 000002e0 29 28 61 78 5e 62 2b 63 29 5e 28 64 2d 31 29 0a |)(ax^b+c)^(d-1).| 000002f0 54 6f 20 64 69 66 66 65 72 65 6e 74 69 61 74 65 |To differentiate| 00000300 20 61 20 62 72 61 63 6b 65 74 20 73 75 63 68 20 | a bracket such | 00000310 61 73 20 74 68 69 73 0a 77 69 74 68 20 72 65 73 |as this.with res| 00000320 70 65 63 74 20 74 6f 20 78 2c 20 79 6f 75 20 77 |pect to x, you w| 00000330 6f 75 6c 64 20 62 72 69 6e 67 20 64 6f 77 6e 0a |ould bring down.| 00000340 74 68 65 20 70 6f 77 65 72 20 28 64 29 2c 20 64 |the power (d), d| 00000350 69 66 66 65 72 65 6e 74 69 61 74 65 20 77 68 61 |ifferentiate wha| 00000360 74 20 69 73 20 69 6e 20 0a 74 68 65 20 62 72 61 |t is in .the bra| 00000370 63 6b 65 74 20 61 6e 64 20 6d 75 6c 74 69 70 6c |cket and multipl| 00000380 79 20 74 68 65 73 65 20 62 79 20 74 68 65 20 0a |y these by the .| 00000390 62 72 61 63 6b 65 74 20 77 69 74 68 20 74 68 65 |bracket with the| 000003a0 20 70 6f 77 65 72 20 72 65 64 75 63 65 64 20 62 | power reduced b| 000003b0 79 20 31 2e 0a 64 79 2f 64 78 3d 28 64 79 2f 64 |y 1..dy/dx=(dy/d| 000003c0 75 29 28 64 75 2f 64 78 29 0a 57 68 65 72 65 20 |u)(du/dx).Where | 000003d0 64 79 2f 64 78 20 69 73 20 74 68 65 20 64 69 66 |dy/dx is the dif| 000003e0 66 65 72 65 6e 74 69 61 6c 20 6f 66 20 74 68 65 |ferential of the| 000003f0 0a 65 71 75 61 74 69 6f 6e 20 79 3d 66 28 78 29 |.equation y=f(x)| 00000400 2e 20 54 68 69 73 20 69 73 20 63 61 6c 6c 65 64 |. This is called| 00000410 20 74 68 65 20 63 68 61 69 6e 0a 72 75 6c 65 20 | the chain.rule | 00000420 61 6e 64 20 69 74 20 69 73 20 6e 6f 74 20 61 73 |and it is not as| 00000430 20 6f 62 76 69 6f 75 73 20 61 73 20 69 74 20 66 | obvious as it f| 00000440 69 72 73 74 0a 73 65 65 6d 73 20 61 73 20 74 68 |irst.seems as th| 00000450 65 79 20 61 72 65 20 4e 4f 54 20 66 72 61 63 74 |ey are NOT fract| 00000460 69 6f 6e 73 2c 20 62 75 74 20 74 68 65 79 0a 63 |ions, but they.c| 00000470 61 6e 20 73 6f 6d 65 74 69 6d 65 73 20 62 65 20 |an sometimes be | 00000480 74 72 65 61 74 65 64 20 61 73 20 69 66 20 74 68 |treated as if th| 00000490 65 79 20 61 72 65 2e 0a 75 76 3e 76 28 64 75 2f |ey are..uv>v(du/| 000004a0 64 78 29 2b 75 28 64 76 2f 64 78 29 0a 54 68 69 |dx)+u(dv/dx).Thi| 000004b0 73 20 69 73 20 63 61 6c 6c 65 64 20 74 68 65 20 |s is called the | 000004c0 70 72 6f 64 75 63 74 20 72 75 6c 65 2c 20 77 68 |product rule, wh| 000004d0 65 72 65 20 75 0a 61 6e 64 20 76 20 61 72 65 20 |ere u.and v are | 000004e0 62 6f 74 68 20 66 75 6e 63 74 69 6f 6e 73 20 6f |both functions o| 000004f0 66 20 78 2e 0a 0a 0a 0a 75 2f 76 3e 28 76 28 64 |f x.....u/v>(v(d| 00000500 75 2f 64 78 29 2d 75 28 64 76 2f 64 78 29 29 2f |u/dx)-u(dv/dx))/| 00000510 76 b2 0a 54 68 69 73 20 69 73 20 63 61 6c 6c 65 |v..This is calle| 00000520 64 20 74 68 65 20 71 75 6f 74 69 65 6e 74 20 72 |d the quotient r| 00000530 75 6c 65 20 77 68 65 72 65 20 75 0a 61 6e 64 20 |ule where u.and | 00000540 76 20 61 72 65 20 62 6f 74 68 20 66 75 6e 63 74 |v are both funct| 00000550 69 6f 6e 73 20 6f 66 20 78 2e 0a 0a 0a 0a 0a 0a |ions of x.......| 00000560 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a |................| * 00000590 0a 0a 0a 0a 0a 0a 0a 0a 0a |.........| 00000599
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