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Normal Density Derivatives

This table list the normal density function and its lower derivatives (to sixth derivative). For higher derivative, see Higher Derivatives of Normal Density Function
Normal Density FunctionDerivative of Normal Density Function

How To Use:

  • You need to find the value of first derivative of the Normal Density Function at Z = 0.02
  • Find the row with Z = 0.02
  • Find the column φ(1)(x), the number inside the cell is 3z79772500.
    See How To Read Numbers in Tables.
  • The result is -0.0079772500
Z φ(x) φ(1)(x) φ(2)(x) φ(3)(x) φ(4)(x) φ(5)(x) φ(6)(x)
0.00.3989422800.39894228001.1968268405.98413421
0.02.3988625003z79772500.398702955.0239285591.19563029.1196268425.97575893
0.04.398623254.015944930.397985457.0478092791.19204400.2389188745.95066324
0.06.398224830.023893490.396791221.0715944531.18607800.3575424915.90893743
0.08.397667706.031813416.395122632.0952366431.17774897.4751664915.85073151
0.10.396952547.039695255.392983022.1186888121.16708018.5914632655.77625460
0.12.396080212.047529625.390376657.1419044501.15410144.7061099715.68577398
0.14.395051741.055307244.387308727.1648377091.13884890.8187896835.57961395
0.16.393868362.063018938.383785332.1874435291.12136503.9291925205.45815435
0.18.392531483.070655667.379813463.2096777571.101698391.037016745.32182895
Zφ(x)φ(1)(x)φ(2)(x) φ(3)(x)φ(4)(x)φ(5)(x)φ(6)(x)
0.20.391042694.078208539.375400986.2314972751.079903501.141969805.17112356
0.22.389403759.085668827.370556617.2528601101.056040631.243769385.00657387
0.24.387616615.093027988.365289898.2737255511.030175561.342144344.82876317
0.26.385683369.100277676.359611173.2940542571.002379411.436835684.63831979
0.28.383606292.107409762.353531559.313808360.9727283361.527597374.43591441
0.30.381387815.114416345.347062912.332951563.9413032671.614197234.22225717
0.32.379030526.121289768.340217800.351449233.9081896461.696417623.99809459
0.34.376537162.128022635.333009466.369268488.8734771121.774056173.76420646
0.36.373910605.134607818.325451791.386378281.8372591921.846926433.52140244
0.38.371153879.141038474.317559259.402749467.7996329801.914858403.27051871
Zφ(x)φ(1)(x)φ(2)(x) φ(3)(x)φ(4)(x)φ(5)(x)φ(6)(x)
0.40.368270140.147308056.309346918.418354879.7606988021.977699043.01241439
0.42.365262673.153410323.300830337.433169387.7205598692.035312692.74796802
0.44.362134882.159339348.292025569.447169947.6793219312.087581442.47807382
0.46.358890291.165089534.282949105.460335656.6370929142.134405372.20363810
0.48.355532529.170655614.273617834.472647788.5939825642.175702781.92557548
0.50.352065327.176032663.264048995.484089824.5501020732.211410331.64480520
0.52.348492513.181216107.254260137.494647485.5055637202.241483071.36224740
0.54.344818001.186201721.244269072.504308741.4604804972.265894431.07881949
0.56.341045789.190985642.234093829.513063828.4149657442.28463613.795432491
0.58.337179944.195564367.223752611.520905249.3691327882.29771801.512987491
Zφ(x)φ(1)(x)φ(2)(x) φ(3)(x)φ(4)(x)φ(5)(x)φ(6)(x)
0.60.333224603.199934762.213263746.527827771.3230945752.30516783.232372177
0.62.329183961.204094056.202645646.533828412.2769633232.30703091.045542547
0.64.325062264.208039849.191916761.538906425.2308501702.30336981.319905826
0.66.320863804.211770110.181095531.543063271.1848648332.29426388.589889990
0.68.316592908.215283177.170200347.546302591.1391152802.27980875.854693552
0.70.312253933.218577753.159249506.548630161.0937074052.260115831.11354405
0.72.307851260.221652908.148261167.550053855.0487447252.235311621.36570074
0.74.303389284.224508070.137253312.5505835913z432807872.205537141.61045709
0.76.298872406.227143028.126243704.550231272.0394446542.170947151.84714311
0.78.294305030.229557923.115249850.549010729.0824788202.131709442.07512746
Zφ(x)φ(1)(x)φ(2)(x) φ(3)(x)φ(4)(x)φ(5)(x)φ(6)(x)
0.80.289691553.231753242.104288959.546937652.1246832442.088004012.29381943
0.82.285036358.233729814.093377911.544029515.1659704692.040022282.50267061
0.84.280343811.235488801.082533218.540305505.2062569711.987966172.70117643
0.86.275618247.237031693.071770992.535786438.2454633621.932047262.88887745
0.88.270863972.238360295.061106912.530494673.2835145761.872485863.06536044
0.90.266085250.239476725.050556197.524454028.3203400321.809510083.23025923
0.92.261286301.240383397.040133576.517689684.3558737821.743354863.38325538
0.94.256471294.241083017.029853259.510228097.3900546351.674261033.52407854
0.96.251644341.241578567.019728916.502096895.4228262701.602474363.65250673
0.98.246809491.2418733013z97736558.493324784.4541373211.528244563.76836628
Zφ(x)φ(1)(x)φ(2)(x) φ(3)(x)φ(4)(x)φ(5)(x)φ(6)(x)
1.00.241970725.2419707250.483941449.4839414491.451824353.87153159
1.02.237131952.2418745913z95801309.473977449.5121973901.373468463.96192478
1.04.232297005.241588885.018955436.463464117.5388689881.293432724.03951497
1.06.227469632.241117810.028115247.452433459.5639252071.211973124.10431754
1.08.222653499.240465779.037049542.440918052.5873401221.129344874.15639308
1.10.217852177.239637395.045748957.428950937.6090929021.045801554.19584622
1.12.213069147.238637444.054204791.416565523.629167758.9615942024.22282430
1.14.208307790.237470881.062409014.403795485.647553895.8769705014.23751585
1.16.203571388.236142810.070354272.390674666.664245427.7921739664.24014894
1.18.198863119.234658481.078033888.377236974.679241293.7074431694.23098941
Zφ(x)φ(1)(x)φ(2)(x) φ(3)(x)φ(4)(x)φ(5)(x)φ(6)(x)
1.20.194186055.233023266.085441864.363516295.692545146.6230110044.21033894
1.22.189543158.231242653.092572878.349546394.704165236.5391039884.17853305
1.24.184937281.229322228.099422282.335360827.714114272.4559416104.13593896
1.26.180371163.227267666.105986096.320992851.722409279.3737357134.08295339
1.28.175847430.225084711.112261000.306475342.729071437.2926899304.02000029
1.30.171368592.222779170.118244329.291840712.734125911.2129991643.94752847
1.32.166937042.220356895.123934060.277120831.737601677.1348491123.86600921
1.34.162555055.217823774.129328802.262346953.739531323.0584158403.77593384
1.36.158224790.215185715.134427782.247549646.739950865.0161345903.67781128
1.38.153948287.212448636.139230831.232758725.738899533.0886464543.57216556
Zφ(x)φ(1)(x)φ(2)(x) φ(3)(x)φ(4)(x)φ(5)(x)φ(6)(x)
1.40.149727466.209618452.143738367.218003190.736419567.1589746343.45953335
1.42.145564130.206701065.147951382.203311167.732556003.2269848553.34046152
1.44.141459965.203702350.151871419.188709857.727356450.2925538603.21550469
1.46.137416539.200628147.155500556.174225483.720870873.3555695423.08522283
1.48.133435304.197484250.158841386.159883249.713151366.4159310262.95017891
1.50.129517596.194276393.161896995.145707295.704251926.4735487092.81093657
1.52.125664637.191010248.164670940.131720667.694228234.5283442482.66805791
1.54.121877537.187691407.167167230.117945280.683137421.5802505072.52210132
1.56.118157295.184325380.169390298.104401895.671037851.6292114672.37361937
1.58.114504800.180917584.171344983.091110096.657988900.6751820802.22315681
Zφ(x)φ(1)(x)φ(2)(x) φ(3)(x)φ(4)(x)φ(5)(x)φ(6)(x)
1.60.110920835.177473335.173036502.078088268.644050734.7181281052.07124870
1.62.107406075.173997842.174470428.065353589.629284100.7580258851.91841857
1.64.103961095.170496196.175652667.052922019.613750112.7948621061.76517671
1.66.100586368.166973372.176589428.040808292.597510050.8286335151.61201862
1.68.097282269.163434212.177287208.029025916.580625162.8593466081.45942351
1.70.094049077.159883432.177752756.017587177.563156470.8870172901.30785296
1.72.090886979.156325604.1779930603z65031451.545164589.9116705121.15774966
1.74.087796071.152765163.1780153133z42163185.526709544.9333398811.00953633
1.76.084776361.149206396.177826895.014562544.507850609.952067248.863614686
1.78.081827776.145653441.177435349.024528040.488646138.967902284.720364625
Zφ(x)φ(1)(x)φ(2)(x) φ(3)(x)φ(4)(x)φ(5)(x)φ(6)(x)
1.80.078950158.142110285.176848355.034106468.469153421.980902031.580143448
1.82.076143274.138580758.176073706.043292629.449428534.991130447.443285256
1.84.073406813.135068535.175119292.052082427.429526210.998657936.310100450
1.86.070740393.131577132.173993072.060472850.4094997151.00356087.180875358
1.88.068143566.128109904.172703054.068461933.3894007281.00592110.055871972
1.90.065615815.124670048.171257277.076048729.3692792441.00582548.064672194
1.92.063156561.121260598.169663787.083233274.3491834731.00336537.180544138
1.94.060765169.117884428.167930621.090016549.329159758.998636126.291555296
1.96.058440944.114544251.166065787.096400442.309252497.991736660.397541369
1.98.056183142.111242621.164077248.102387708.289504080.982768912.498362045
Zφ(x)φ(1)(x)φ(2)(x) φ(3)(x)φ(4)(x)φ(5)(x)φ(6)(x)
2.00.053990967.107981933.161972900.107981933.269954833.971837397.593900632
2.02.051863577.104764425.159760562.113187485.250642966.959048730.684063605
2.04.049800088.101592179.157447957.118009475.231604543.944511168.768780068
2.06.047799575.098467124.155042701.122453716.212873449.928334168.848001141
2.08.045861076.095391039.152552284.126526674.194481371.910627947.921699273
2.10.043983596.092365552.149984062.130235428.176457789.891503067.989867497
2.12.042166107.089392147.147345244.133587624.158829969.8710700321.05251862
2.14.040407554.086472165.144642880.136591432.141622975.8494388951.10968436
2.16.038706856.083606809.141883852.139255502.124859672.8267188981.16141446
2.18.037062910.080797144.139074864.141588916.108560757.8030181131.20777570
Zφ(x)φ(1)(x)φ(2)(x) φ(3)(x)φ(4)(x)φ(5)(x)φ(6)(x)
2.20.035474593.078044104.136222437.143601152.092744776.7784431141.24885097
2.22.033940763.075348494.133332894.145302036.077428162.7530986641.28473823
2.24.032460266.072710995.130412363.146701704.062625274.7270874271.31554947
2.26.031031932.070132167.127466765.147810555.048348441.7005096951.34140971
2.28.029654585.067612453.124501809.148639218.034608011.6734631351.36245589
2.30.028327038.065152187.121522992.149198508.021412408.6460425691.37883587
2.32.027048100.062751591.118535591.1494993903z87681889.6183397591.39070730
2.34.025816575.060410787.115544665.1495529433z33198918.5904432261.39823661
2.36.024631269.058129796.112555048.149370323.014848817.5624380831.40159796
2.38.023490985.055908545.109571352.148962728.025817236.5344058901.40097220
Zφ(x)φ(1)(x)φ(2)(x) φ(3)(x)φ(4)(x)φ(5)(x)φ(6)(x)
2.40.022394530.053746873.106597964.148341369.036225392.5064245331.39654584
2.42.021340715.051644530.103639048.147517436.046075051.4785681201.38851010
2.44.020328356.049601188.100698543.146502069.055369419.4509068931.37705991
2.46.019356277.047616441.097780168.145306331.064113071.4235071691.36239299
2.48.018423311.045689810.094887419.143941179.072311866.3964312881.34470892
2.50.017528300.043820751.092023578.142417442.079972871.3697375891.32420833
2.52.016670101.042008654.089191708.140745795.087104280.3434803931.30109199
2.54.015847579.040252851.086394662.138936740.093715333.3177100121.27556010
2.56.015059616.038552618.083635085.137000583.099816236.2924727671.24781146
2.58.014305109.036907181.080915419.134947417.105418081.2678110201.21804284
Zφ(x)φ(1)(x)φ(2)(x) φ(3)(x)φ(4)(x)φ(5)(x)φ(6)(x)
2.60.013582969.035315720.078237903.132787107.110532770.2437632261.18644824
2.62.012892126.033777370.075604584.130529270.115172935.2203639911.15321833
2.64.012231526.032291230.073017320.128183265.119351860.1976441491.11853985
2.66.011600135.030856359.070477781.125758178.123083412.1756308381.08259509
2.68.010996937.029471790.067987461.123262815.126381962.1543476041.04556139
2.70.010420935.028136524.065547680.120705688.129262318.1338144941.00761072
2.723z98711538.026849538.063159590.118095009.131739654.114048178.968909315
2.743z93466384.025609789.060824184.115438685.133829447.095062058.929617272
2.763z88464544.024416214.058542297.112744310.135547407.076866399.889888295
2.783z83696892.023267736.056314617.110019162.136909421.059468457.849869418
Zφ(x)φ(1)(x)φ(2)(x) φ(3)(x)φ(4)(x)φ(5)(x)φ(6)(x)
2.803z79154516.022163264.054141689.107270200.137931493.042872619.809700798
2.823z74828725.021101701.052023923.104504062.138629685.027080535.769515533
2.843z70711049.020081938.049961599.101727065.139020067.012091267.729439535
2.863z66793237.019102866.047954873.098945204.1391186663z20985675.689591426
2.883z63067264.018163372.046003785.096164157.138941417.015494652.650082484
2.903z59525324.017262344.044108265.093389281.138504120.028104822.611016613
2.923z56159836.016398672.042268139.090625621.137822398.039938916.572490355
2.943z52963439.015571251.040483134.087877912.136911659.051008630.534592923
2.963z49928992.014778982.038752887.085150581.135787060.061327373.497406273
2.983z47049575.014020773.037076947.082447756.134463471.070910120.461005199
Zφ(x)φ(1)(x)φ(2)(x) φ(3)(x)φ(4)(x)φ(5)(x)φ(6)(x)
3.003z44318484.013295545.035454787.079773271.132955452.079773271.425457448
3.023z41729230.012602227.033885804.077130673.131277220.087934514.390823868
3.043z39275536.011939763.032369326.074523225.129442626.095412683.357158573
3.063z36951337.011307109.030904620.071953920.127465134.102227630.324509123
3.083z34750774.010703238.029490897.069425485.125357804.108400096.292916724
3.103z32668191.010127139.028127312.066940389.123133271.113951582.262416450
3.123z306981333z95778175.026812977.064500854.120803733.118904231.233037466
3.143z288353433z90542979.025546961.062108862.118380943.123280713.204803273
3.163z270747583z85556234.024328294.059766163.115876192.127104115.177731955
3.183z254115003z80808571.023155976.057474288.113300309.130397831.151836442
Zφ(x)φ(1)(x)φ(2)(x) φ(3)(x)φ(4)(x)φ(5)(x)φ(6)(x)
3.203z238408823z76290822.022028975.055234555.110663653.133185466.127124771
3.223z223583943z71994030.020946238.053048081.107976106.135490738.103600355
3.243z209597063z67909448.019906690.050915788.105247080.137337390.081262258
3.263z196406593z64028547.018909241.048838415.102485510.138749105.060105470
3.283z183972623z60343019.017952784.046816528.099699859.139749426.040121176
3.303z172256893z56844775.017036207.044850527.096898120.140361687.021297034
3.323z161222753z53525952.016158389.042940660.094087825.1406089393z36174468
3.343z150835063z50378911.015318206.041087024.091276045.140513892.012936176
3.363z141060233z47396236.014514533.039289584.088469402.140098857.028385147
3.383z131866093z44570738.013746249.037548173.085674078.139385692.042753251
Zφ(x)φ(1)(x)φ(2)(x) φ(3)(x)φ(4)(x)φ(5)(x)φ(6)(x)
3.403z123221923z41895452.013012234.035862507.082895819.138395759.056066485
3.423z115098343z39363634.012311379.034232190.080139954.137149879.068352819
3.443z107467333z36968763.011642581.032656726.077411396.135668295.079641957
3.463z100302133z34704537.011004749.031135522.074714662.133970641.089965108
3.484z935772163z32564871.010396803.029667900.072053884.132075914.099354764
3.504z872682703z305438943z98176803.028253102.069432817.130002450.107844491
3.524z813521233z286359473z92663322.026890300.066854859.127767905.115468728
3.544z758067193z268355793z87417276.025578600.064323061.125389237.122262592
3.564z706110703z251375413z82428539.024317052.061840142.122882699.128261699
3.584z657452183z235367883z77687179.023104652.059408502.120263828.133501993
Zφ(x)φ(1)(x)φ(2)(x) φ(3)(x)φ(4)(x)φ(5)(x)φ(6)(x)
3.604z611901933z220284693z73183471.021940356.057030239.117547438.138019581
3.624z569279783z206079283z68907902.020823075.054707161.114747622.141850588
3.644z529414683z192706943z64851181.019751691.052440801.111877751.145031010
3.664z492144333z180124823z61004242.018725056.050232433.108950480.147596592
3.684z457314813z168291853z57358253.017742000.048083085.105977751.149582700
3.704z424780273z157168703z53904616.016801334.045993551.102970803.151024215
3.724z394402523z146717743z50634974.015901855.043964410.099940184.151955434
3.744z366050763z136902993z47541209.015042352.041996036.096895763.152409976
3.764z339601213z127690063z44615449.014221608.040088610.093846744.152420705
3.784z314936813z119046123z41850063.013438402.038242139.090801680.152019653
Zφ(x)φ(1)(x)φ(2)(x) φ(3)(x)φ(4)(x)φ(5)(x)φ(6)(x)
3.804z291946933z110939833z39237667.012691517.036456464.087768495.151237962
3.824z270527033z103341333z36771116.011979740.034731271.084754497.150105823
3.844z250578444z962221233z34443511.011301866.033066111.081766403.148652434
3.864z232008044z895551033z32248189.010656699.031460402.078810354.146905958
3.884z214727984z833144573z30178729.010043058.029913446.075891938.144893491
3.904z198655474z774756343z282289423z94597749.028424439.073016214.142641037
3.924z183712504z720152993z263928723z89057000.026992482.070187730.140173492
3.944z169825604z669112863z246647913z83797018.025616588.067410549.137514625
3.964z156925634z621425513z230391943z78806698.024295694.064688270.134687077
3.984z144947564z576891293z215107983z74075149.023028670.062024047.131712358
Zφ(x)φ(1)(x)φ(2)(x) φ(3)(x)φ(4)(x)φ(5)(x)φ(6)(x)
4.004z133830234z535320903z200745343z69591717.021814327.059420620.128610847
4.024z123516164z496534983z187255443z65345989.020651424.056880330.125401805
4.044z113951404z460363653z174591773z61327804.019538679.054405144.122103383
4.064z105085264z426646163z162709813z57527261.018474774.051996677.118732639
4.085z968702084z395230453z151567003z53934728.017458359.049656213.115305556
4.105z892616574z365972793z141122683z50540843.016488065.047384730.111837068
4.125z822178174z338737403z131338033z47336520.015562505.045182914.108341079
4.145z756995364z313396083z122176023z44312952.014680281.043051184.104830495
4.165z696701544z289827843z113601373z41461612.013839989.040989712.101317253
4.185z640953634z267918623z105580453z38774254.013040225.038998438.097812347
Zφ(x)φ(1)(x)φ(2)(x) φ(3)(x)φ(4)(x)φ(5)(x)φ(6)(x)
4.205z589430684z247560884z980812653z36242914.012279586.037077095.094325869
4.225z541832614z228653364z910733933z33859904.011556678.035225219.090867034
4.245z497879024z211100704z845279093z31617819.010870118.033442173.087444223
4.265z457308004z194813214z784173463z29509525.010218537.031727159.084065011
4.285z419875034z179706514z727156373z275281623z96005844.030079236.080736209
4.305z385351974z165701354z673980593z256671383z90149278.028497334.077463898
4.325z353526034z152723244z624411823z239201263z84602588.026980268.074253463
4.345z324198874z140702314z578228133z222810553z79352934.025526751.071109634
4.365z297185684z129572964z535219523z207441123z74387742.024135411.068036520
4.385z272314364z119273694z495187323z193037313z69694721.022804796.065037644
Zφ(x)φ(1)(x)φ(2)(x) φ(3)(x)φ(4)(x)φ(5)(x)φ(6)(x)
4.405z249424714z109746874z457943773z179545893z65261876.021533390.062115979
4.425z228367724z100938534z423311533z166915993z61077522.020319625.059273982
4.445z209004785z927981224z391123183z155099073z57130291.019161887.056513631
4.465z191207105z852783654z361220803z144048803z53409142.018058525.053836452
4.485z174855015z783350454z333455503z133721053z49903367.017007866.051243558
4.505z159837415z719268354z307687023z124073793z46602595.016008216.048735675
4.525z146051185z660151344z283783293z115067023z43496794.015057870.046313176
4.545z133400665z605639024z261620053z106662723z40576274.014155120.043976106
4.565z121797175z555395094z241080454z988247813z37831687.013298258.041724213
4.585z111158505z509105934z222054674z915189193z35254025.012485587.039556974
Zφ(x)φ(1)(x)φ(2)(x) φ(3)(x)φ(4)(x)φ(5)(x)φ(6)(x)
4.605z101408525z466479204z204439584z847126223z32834619.011715420.037473621
4.626z924767375z427242524z188138374z783750773z30565135.010986089.035473164
4.646z842979145z391142324z173060254z724771073z28437570.010295948.033554415
4.666z768117115z357942574z159120074z669911003z264442513z96433768.031716011
4.686z699623415z327423764z146238084z618909483z245778213z90267824.029956431
4.706z636982525z299381784z134339614z571519823z228312433z84446051.028274022
4.726z579718255z273627014z123354774z527509103z211977873z78953189.026667012
4.746z527391005z249983334z113218194z486657563z196710223z73774344.025133528
4.766z479595105z228287274z103868794z448757983z182448163z68895006.023671615
4.786z435956395z208387155z952494954z413615163z169133203z64301062.022279248
Zφ(x)φ(1)(x)φ(2)(x) φ(3)(x)φ(4)(x)φ(5)(x)φ(6)(x)
4.806z396129915z190142365z873070324z381045283z156709633z59978809.020954347
4.826z359797805z173422545z799916864z350875423z145124453z55914966.019694792
4.846z326667305z158106975z732571014z322942983z134327273z52096680.018498429
4.866z296468875z144083875z670600724z297095183z124270233z48511527.017363090
4.886z268954505z131249795z613603554z273188573z114907923z45147521.016286594
4.906z243896075z119509085z561204874z251088573z106197253z41993111.015266762
4.926z221083895z108773275z513056114z230668954z980974413z39037183.014301422
4.946z200325236z989606655z468833164z211811454z905698613z36269053.013388419
4.966z181443126z899957875z428234794z194405304z835779853z33678468.012525621
4.986z164275066z818089795z390981214z178346854z770872943z31255598.011710923
5.006z148671956z743359765z356812684z163539154z710651933z28991031.010942256


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