Home » Archimedes archive » Acorn User » AU 1997-01 B.adf » Regulars » StarInfo/Allen/!Ignotum/Formulae/Formulae/Diff
StarInfo/Allen/!Ignotum/Formulae/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 » Acorn User » AU 1997-01 B.adf » Regulars |
| Filename: | StarInfo/Allen/!Ignotum/Formulae/Formulae/Diff |
| Read OK: | ✔ |
| File size: | 071D bytes |
| Load address: | 0000 |
| Exec address: | 0000 |
File contents
# Maths > Differentiation # New entries should take the form of: # Formula # Note 1 # Note 2 # Note 3 # Note 4 # Note 5 # Formula # Note 1 # And so on... # To fit snugly into the window, each line should be no longer # than 42 characters. # There is a limit of 25 formulas per topic. # Any notes should be made here, at the beginning and should # be preceeded by a hash (#). 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 0a 23 20 4e 65 77 |entiation..# New| 00000020 20 65 6e 74 72 69 65 73 20 73 68 6f 75 6c 64 20 | entries should | 00000030 74 61 6b 65 20 74 68 65 20 66 6f 72 6d 20 6f 66 |take the form of| 00000040 3a 0a 23 20 20 20 20 20 46 6f 72 6d 75 6c 61 0a |:.# Formula.| 00000050 23 20 20 20 20 20 4e 6f 74 65 20 31 0a 23 20 20 |# Note 1.# | 00000060 20 20 20 4e 6f 74 65 20 32 0a 23 20 20 20 20 20 | Note 2.# | 00000070 4e 6f 74 65 20 33 0a 23 20 20 20 20 20 4e 6f 74 |Note 3.# Not| 00000080 65 20 34 0a 23 20 20 20 20 20 4e 6f 74 65 20 35 |e 4.# Note 5| 00000090 0a 23 20 20 20 20 20 46 6f 72 6d 75 6c 61 0a 23 |.# Formula.#| 000000a0 20 20 20 20 20 4e 6f 74 65 20 31 0a 23 20 20 20 | Note 1.# | 000000b0 20 20 41 6e 64 20 73 6f 20 6f 6e 2e 2e 2e 0a 23 | And so on....#| 000000c0 20 54 6f 20 66 69 74 20 73 6e 75 67 6c 79 20 69 | To fit snugly i| 000000d0 6e 74 6f 20 74 68 65 20 77 69 6e 64 6f 77 2c 20 |nto the window, | 000000e0 65 61 63 68 20 6c 69 6e 65 20 73 68 6f 75 6c 64 |each line should| 000000f0 20 62 65 20 6e 6f 20 6c 6f 6e 67 65 72 0a 23 20 | be no longer.# | 00000100 74 68 61 6e 20 34 32 20 63 68 61 72 61 63 74 65 |than 42 characte| 00000110 72 73 2e 0a 23 20 54 68 65 72 65 20 69 73 20 61 |rs..# There is a| 00000120 20 6c 69 6d 69 74 20 6f 66 20 32 35 20 66 6f 72 | limit of 25 for| 00000130 6d 75 6c 61 73 20 70 65 72 20 74 6f 70 69 63 2e |mulas per topic.| 00000140 0a 0a 23 20 41 6e 79 20 6e 6f 74 65 73 20 73 68 |..# Any notes sh| 00000150 6f 75 6c 64 20 62 65 20 6d 61 64 65 20 68 65 72 |ould be made her| 00000160 65 2c 20 61 74 20 74 68 65 20 62 65 67 69 6e 6e |e, at the beginn| 00000170 69 6e 67 20 61 6e 64 20 73 68 6f 75 6c 64 0a 23 |ing and should.#| 00000180 20 62 65 20 70 72 65 63 65 65 64 65 64 20 62 79 | be preceeded by| 00000190 20 61 20 68 61 73 68 20 28 23 29 2e 0a 0a 73 69 | a hash (#)...si| 000001a0 6e 78 3e 63 6f 73 78 0a 27 3e 27 20 6d 65 61 6e |nx>cosx.'>' mean| 000001b0 73 20 27 67 6f 65 73 20 74 6f 27 20 72 61 74 68 |s 'goes to' rath| 000001c0 65 72 20 74 68 61 6e 20 27 67 72 65 61 74 65 72 |er than 'greater| 000001d0 0a 74 68 61 6e 27 2e 0a 0a 0a 0a 63 6f 73 78 3e |.than'.....cosx>| 000001e0 2d 73 69 6e 78 0a 0a 0a 0a 0a 0a 74 61 6e 78 3e |-sinx......tanx>| 000001f0 73 65 63 b2 78 0a 0a 0a 0a 0a 0a 73 65 63 78 3e |sec.x......secx>| 00000200 73 65 63 78 74 61 6e 78 0a 0a 0a 0a 0a 0a 63 6f |secxtanx......co| 00000210 73 65 63 78 3e 2d 63 6f 73 65 63 78 63 6f 74 78 |secx>-cosecxcotx| 00000220 0a 0a 0a 0a 0a 0a 63 6f 74 78 3e 2d 63 6f 73 65 |......cotx>-cose| 00000230 63 b2 78 0a 0a 0a 0a 0a 0a 65 5e 78 3e 65 5e 78 |c.x......e^x>e^x| 00000240 0a 57 68 65 72 65 20 65 20 69 73 20 74 68 65 20 |.Where e is the | 00000250 63 6f 6e 73 74 61 6e 74 20 32 2e 37 31 38 2e 20 |constant 2.718. | 00000260 54 68 69 73 20 64 6f 65 73 0a 6e 6f 74 20 63 68 |This does.not ch| 00000270 61 6e 67 65 20 77 68 65 6e 20 64 69 66 66 65 72 |ange when differ| 00000280 65 6e 74 69 61 74 65 64 2e 20 48 6f 77 65 76 65 |entiated. Howeve| 00000290 72 2c 0a 65 5e 61 78 3e 61 65 5e 61 78 2e 0a 0a |r,.e^ax>ae^ax...| 000002a0 0a 61 72 63 73 69 6e 78 3e 31 2f 28 31 2d 78 b2 |.arcsinx>1/(1-x.| 000002b0 29 5e bd 0a 41 72 63 73 69 6e 65 3d 69 6e 76 65 |)^..Arcsine=inve| 000002c0 72 73 65 20 73 69 6e 65 2c 20 61 6c 73 6f 20 77 |rse sine, also w| 000002d0 72 69 74 74 65 6e 20 61 73 0a 73 69 6e 5e 28 2d |ritten as.sin^(-| 000002e0 31 29 78 2e 20 54 68 69 73 20 69 73 20 6e 6f 74 |1)x. This is not| 000002f0 20 74 68 65 20 73 61 6d 65 20 61 73 20 31 2f 73 | the same as 1/s| 00000300 69 6e 78 0a 77 68 69 63 68 20 77 6f 75 6c 64 20 |inx.which would | 00000310 62 65 20 77 72 69 74 74 65 6e 20 61 73 20 28 73 |be written as (s| 00000320 69 6e 78 29 5e 2d 31 2e 0a 0a 0a 61 72 63 63 6f |inx)^-1....arcco| 00000330 73 78 3e 2d 31 2f 28 31 2d 78 b2 29 5e bd 0a 41 |sx>-1/(1-x.)^..A| 00000340 72 63 63 6f 73 69 6e 65 3d 69 6e 76 65 72 73 65 |rccosine=inverse| 00000350 20 63 6f 73 69 6e 65 2c 20 61 6c 73 6f 20 77 72 | cosine, also wr| 00000360 69 74 74 65 6e 20 61 73 0a 63 6f 73 5e 28 2d 31 |itten as.cos^(-1| 00000370 29 78 2e 20 54 68 69 73 20 69 73 20 6e 6f 74 20 |)x. This is not | 00000380 74 68 65 20 73 61 6d 65 20 61 73 20 31 2f 63 6f |the same as 1/co| 00000390 73 78 0a 77 68 69 63 68 20 77 6f 75 6c 64 20 62 |sx.which would b| 000003a0 65 20 77 72 69 74 74 65 6e 20 61 73 20 28 63 6f |e written as (co| 000003b0 73 78 29 5e 2d 31 2e 0a 0a 0a 61 72 63 74 61 6e |sx)^-1....arctan| 000003c0 78 3e 31 2f 28 31 2b 78 b2 29 0a 41 72 63 74 61 |x>1/(1+x.).Arcta| 000003d0 6e 3d 69 6e 76 65 72 73 65 20 74 61 6e 2c 20 61 |n=inverse tan, a| 000003e0 6c 73 6f 20 77 72 69 74 74 65 6e 20 61 73 0a 74 |lso written as.t| 000003f0 61 6e 5e 28 2d 31 29 78 2e 20 54 68 69 73 20 69 |an^(-1)x. This i| 00000400 73 20 6e 6f 74 20 74 68 65 20 73 61 6d 65 20 61 |s not the same a| 00000410 73 20 31 2f 74 61 6e 78 0a 77 68 69 63 68 20 77 |s 1/tanx.which w| 00000420 6f 75 6c 64 20 62 65 20 77 72 69 74 74 65 6e 20 |ould be written | 00000430 61 73 20 28 74 61 6e 78 29 5e 2d 31 2e 0a 0a 0a |as (tanx)^-1....| 00000440 6c 6f 67 78 3e 31 2f 78 0a 0a 0a 0a 0a 0a 28 61 |logx>1/x......(a| 00000450 78 5e 62 2b 63 29 5e 64 3e 64 28 61 62 78 5e 28 |x^b+c)^d>d(abx^(| 00000460 62 2d 31 29 29 28 61 78 5e 62 2b 63 29 5e 28 64 |b-1))(ax^b+c)^(d| 00000470 2d 31 29 0a 54 6f 20 64 69 66 66 65 72 65 6e 74 |-1).To different| 00000480 69 61 74 65 20 61 20 62 72 61 63 6b 65 74 20 73 |iate a bracket s| 00000490 75 63 68 20 61 73 20 74 68 69 73 0a 77 69 74 68 |uch as this.with| 000004a0 20 72 65 73 70 65 63 74 20 74 6f 20 78 2c 20 79 | respect to x, y| 000004b0 6f 75 20 77 6f 75 6c 64 20 62 72 69 6e 67 20 64 |ou would bring d| 000004c0 6f 77 6e 0a 74 68 65 20 70 6f 77 65 72 20 28 64 |own.the power (d| 000004d0 29 2c 20 64 69 66 66 65 72 65 6e 74 69 61 74 65 |), differentiate| 000004e0 20 77 68 61 74 20 69 73 20 69 6e 20 0a 74 68 65 | what is in .the| 000004f0 20 62 72 61 63 6b 65 74 20 61 6e 64 20 6d 75 6c | bracket and mul| 00000500 74 69 70 6c 79 20 74 68 65 73 65 20 62 79 20 74 |tiply these by t| 00000510 68 65 20 0a 62 72 61 63 6b 65 74 20 77 69 74 68 |he .bracket with| 00000520 20 74 68 65 20 70 6f 77 65 72 20 72 65 64 75 63 | the power reduc| 00000530 65 64 20 62 79 20 31 2e 0a 64 79 2f 64 78 3d 28 |ed by 1..dy/dx=(| 00000540 64 79 2f 64 75 29 28 64 75 2f 64 78 29 0a 57 68 |dy/du)(du/dx).Wh| 00000550 65 72 65 20 64 79 2f 64 78 20 69 73 20 74 68 65 |ere dy/dx is the| 00000560 20 64 69 66 66 65 72 65 6e 74 69 61 6c 20 6f 66 | differential of| 00000570 20 74 68 65 0a 65 71 75 61 74 69 6f 6e 20 79 3d | the.equation y=| 00000580 66 28 78 29 2e 20 54 68 69 73 20 69 73 20 63 61 |f(x). This is ca| 00000590 6c 6c 65 64 20 74 68 65 20 63 68 61 69 6e 0a 72 |lled the chain.r| 000005a0 75 6c 65 20 61 6e 64 20 69 74 20 69 73 20 6e 6f |ule and it is no| 000005b0 74 20 61 73 20 6f 62 76 69 6f 75 73 20 61 73 20 |t as obvious as | 000005c0 69 74 20 66 69 72 73 74 0a 73 65 65 6d 73 20 61 |it first.seems a| 000005d0 73 20 74 68 65 79 20 61 72 65 20 4e 4f 54 20 66 |s they are NOT f| 000005e0 72 61 63 74 69 6f 6e 73 2c 20 62 75 74 20 74 68 |ractions, but th| 000005f0 65 79 0a 63 61 6e 20 73 6f 6d 65 74 69 6d 65 73 |ey.can sometimes| 00000600 20 62 65 20 74 72 65 61 74 65 64 20 61 73 20 69 | be treated as i| 00000610 66 20 74 68 65 79 20 61 72 65 2e 0a 75 76 3e 76 |f they are..uv>v| 00000620 28 64 75 2f 64 78 29 2b 75 28 64 76 2f 64 78 29 |(du/dx)+u(dv/dx)| 00000630 0a 54 68 69 73 20 69 73 20 63 61 6c 6c 65 64 20 |.This is called | 00000640 74 68 65 20 70 72 6f 64 75 63 74 20 72 75 6c 65 |the product rule| 00000650 2c 20 77 68 65 72 65 20 75 0a 61 6e 64 20 76 20 |, where u.and v | 00000660 61 72 65 20 62 6f 74 68 20 66 75 6e 63 74 69 6f |are both functio| 00000670 6e 73 20 6f 66 20 78 2e 0a 0a 0a 0a 75 2f 76 3e |ns of x.....u/v>| 00000680 28 76 28 64 75 2f 64 78 29 2d 75 28 64 76 2f 64 |(v(du/dx)-u(dv/d| 00000690 78 29 29 2f 76 b2 0a 54 68 69 73 20 69 73 20 63 |x))/v..This is c| 000006a0 61 6c 6c 65 64 20 74 68 65 20 71 75 6f 74 69 65 |alled the quotie| 000006b0 6e 74 20 72 75 6c 65 20 77 68 65 72 65 20 75 0a |nt rule where u.| 000006c0 61 6e 64 20 76 20 61 72 65 20 62 6f 74 68 20 66 |and v are both f| 000006d0 75 6e 63 74 69 6f 6e 73 20 6f 66 20 78 2e 0a 0a |unctions of x...| 000006e0 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a |................| * 00000710 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a 0a |.............| 0000071d
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