# Derivative of an interpolating function

I have an interpolating function (which comes from an integral that cannot be done analytically), it looks nice and smooth:

However, since what I need is the derivative of this function, it is not anywhere as nice, and creates even more problems because I need to take another derivative:

What can I do to fix this? Would a finer interpolation (more points) help me get a nice and smooth derivative?

Edit: here is the Data:

Data = {{0., 1.1851102951164046911408964858885408784810.150144072619367},\
{0.01, 1.1851091514138187300211900875520243620510.150144024011082},\
{0.02, 1.1851143532206184764803521392104207945410.150144027484796},\
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{2.23, 1.1805143366353061327931657766518813245810.150319943291501}, \
{2.24, 1.1805056194818603519007060130640351489310.150320013339043}, \
{2.25, 1.1804873579082363689818857724263665991310.150319936232428}, \
{2.2600000000000002,
1.1804735825687032314478120536490515079910.1503200024423}, \
{2.27, 1.1804528442583600361018381960231098211110.15032006948038}, \
{2.2800000000000002,
1.1804331842356967828043292153477249502310.150320097854435}, \
{2.29, 1.1804220549059534064031604986895143354410.150320139600492}, \
{2.3000000000000003,
1.180408066433012891684185873395396206410.150320174993}, {2.31,
1.1803894152928383161390722316896514205410.150320177542772}, \
{2.32, 1.1803845541016595135958639540485456831310.150320276480821}, \
{2.33, 1.1803671893557390232967213957513582798510.150320223681874}, \
{2.34, 1.1803577505356804494646439648304830229610.150320335258167}, \
{2.35, 1.1803423019713492981798107576758516946110.150320348473715}, \
{2.36, 1.1803305697053584093140107296302684737810.150320387547922}, \
{2.37, 1.1803171066200417476056961768902182400610.150320413705433}, \
{2.38, 1.1802985023168703327275067617459632443310.150320399454788}, \
{2.39, 1.1802931561361253834237221027233542695310.150320616336812}, \
{2.4, 1.1802713601638888660902410841954605205810.15032049953037}, \
{2.41, 1.180268375739822780415233211643848056110.150320532384328}, \
{2.42, 1.1802481823394973505239859206026606948110.150320539818848}, \
{2.43, 1.1802406618504339734525642382598168144710.15032057611617}, \
{2.44, 1.1802243226700080408423326081071812959910.150320559046884}, \
{2.45, 1.1802111632755240636134601666587926582110.150320563812805}, \
{2.46, 1.1801993628555082330263280808889801606610.150320637244697}, \
{2.47, 1.1801910158845130900120951985942572098510.150320637647223}, \
{2.48, 1.1801878609633849944985379190546573975310.150320636263197}, \
{2.49, 1.1801645944810930881832702240091075332310.150320604856626}, \
{2.5, 1.1801551931344098641525031396018300474710.15032074122922}, \
{2.5100000000000002,
1.180139509339490710972511947704677564110.150320673420158}, \
{2.52, 1.180152118910321541330366322764536395310.150320879595276}, \
{2.5300000000000002,
1.180130797260289969535658079490307522910.150320703866537}, \
{2.54, 1.180121903257206404663754889843708435310.15032077822856}, \
{2.5500000000000003,
1.1801077832863093442550496398564398183810.150320780679827}, \
{2.56, 1.180098147613293568523492796463426070810.150320797194558}, \
{2.57, 1.180085288088074159823329771569387869110.150320812448818}, \
{2.58, 1.180080336736104895709867924405827942310.150320835462551}, \
{2.59, 1.1800713199807071349303156327540652163210.150320850568097}, \
{2.6, 1.1800620924067932408735493015784579527710.150320869927336}, \
{2.61, 1.1800538897806712811012416552344360943210.150320887923487}, \
{2.62, 1.1800450753020899555618454201879640557910.150320901991472}, \
{2.63, 1.1800368210683696381706015263706643601510.150320917134115}, \
{2.64, 1.1800291801441791135400270792793212568210.150320936635978}, \
{2.65, 1.1800192847400626157568983256246501751110.150320951961788}, \
{2.66, 1.1800130294281492230681948990918706875610.150320969148144}, \
{2.67, 1.1800058584532336859561062379751002453710.150320986058905}, \
{2.68, 1.1799983031939807763395501237185322069210.150320961943892}, \
{2.69, 1.1799883734717350647490823556012090090710.150321004684509}, \
{2.7, 1.1799843585581201836185927988699555474810.150321018737687}, \
{2.71, 1.1799776853807166236517215679607379823510.150321031963806}, \
{2.72, 1.1799708474359630833627767460755072454910.15032104801872}, \
{2.73, 1.1799642399538068246627403101675850421510.150321062210065}, \
{2.74, 1.1799592906644221478627442713109372076110.150321072884452}, \
{2.75, 1.1799523017023794965181267114757958164610.15032108285379}, \
{2.7600000000000002,
1.1799461284332631973575710473906372862510.150321099861214}, \
{2.77, 1.1799388250510726587831485207059785544510.150321086933197}, \
{2.7800000000000002,
1.1799345264047084511618801474616454619810.150321116360066}, \
{2.79, 1.1799290987080167069484728732262733665810.150321131596627}, \
{2.8000000000000003,
1.1799243912362057218108023273929117175410.150321147110823}, \
{2.81, 1.1799191246773705416957107432204580606810.15032114225105}, \
{2.82, 1.1799148094032442255322618739394575697410.150321154100451}, \
{2.83, 1.1799095887466626727477444752456297554310.150321181093055}, \
{2.84, 1.1799056419420369609477694373516293873510.150321172747566}, \
{2.85, 1.1798999492761816491645293855267872526610.15032120730178}, \
{2.86, 1.1798975550130912395135572896186591318810.150321198361539}, \
{2.87, 1.1798923275435300317151089898389679953510.150321194894682}, \
{2.88, 1.1798884556896132699256499264211901205310.150321197127615}, \
{2.89, 1.1798858974271980358497373065073826099510.150321212434863}, \
{2.9, 1.1798817658189378096242180330055215178810.15032119417042}, \
{2.91, 1.1798787541793795958561398463985356156910.150321197195288}, \
{2.92, 1.1798796593895775444960084766466101037610.150321205153638}, \
{2.93, 1.1798740261340945845951518980665308217210.150321201971883}, \
{2.94, 1.1798722253319229431117701536862997392910.150321206144982}, \
{2.95, 1.1798695734700250159091169422383735435910.150321212486556}, \
{2.96, 1.1798667753560505382310445495353417134110.15032121854462}, \
{2.97, 1.1798632371067419782172839219202379370710.150321215599114}, \
{2.98, 1.1798594964012944274722709585970350911710.150321207294247}, \
{2.99, 1.1798555171366481521741582361154488708410.15032118932838}, \
{3., 1.1798514948077990747118209143327693259110.150321160381248}, \
{3.0100000000000002,
1.1798477013688824351755191698175605372510.150321118990874}, \
{3.02, 1.1798442432688657791413668716265996542310.150321065700378}, \
{3.0300000000000002,
1.1798406786349394151137058322567256814110.150321025113692}, \
{3.04, 1.1798397228071436616792044800640872613410.150320931432793}, \
{3.0500000000000003,
1.1798371309969852336905517211794356025310.150320881950467}, \
{3.06, 1.1798380567222208804250947817474849544310.150320777479246}, \
{3.0700000000000003,
1.1798385924835933924467803500600965181210.150320698649798}, \
{3.08, 1.1798401027891659309757343881586690519410.15032062546959}, \
{3.09, 1.1798409011746254212424479168779007627910.150320559809153},
{3.1,
1.1798299723992470725583563515890581498110.150320733070526}, \
{3.11, 1.1798439305823633840299858003774243730710.150320451743152}, \
{3.12, 1.179845885488646865771926420618724062310.150320429716823}, \
{3.13, 1.1798462453397561521975435868675734856710.150320394862455}, \
{3.14, 1.1798466413333354772643951351660618683510.150320383670744}};


Sorry that this became so long, it was the quickest way to share the data with you.

• Could you provide the code that generated the interpolating function? Jun 27, 2018 at 16:56
• I simply used Func = Interpolation[Data]; Jun 27, 2018 at 17:26
• But what is Data? Jun 27, 2018 at 17:27
• Most likely your problem is sampling too fine. That causes the derivative to expose round off error in the samples. Jun 27, 2018 at 17:34
• @Gordon Hard to say without the Data. Jun 27, 2018 at 18:04

There are probably better ways to do it, but for lack of a better answer, a low pass filter will do it.

Interpolation[Data]
Interpolation[Transpose[{Most@#[[1]], LowpassFilter[Differences@#[[2]], .1]} &@ Transpose[ata]]]
Plot[%%[x], {x, 0, Pi}]
Plot[%%[x], {x, 0, Pi}]


Play around with the cutoff for better results.

... or use a moving average:

Interpolation[MovingAverage[Data, 15]]
Plot[%[x], {x, 0, \[Pi]}]
Plot[%%'[x], {x, 0, \[Pi]}]


• Thank you! Could you perhaps mention what better methods you have in mind, so I could investigate? Jun 27, 2018 at 21:14
• Note: I'm lazy and didn't divide by $\Delta x$; this would only change the scale of the graph, but not its overall shape. Jun 27, 2018 at 21:15
• @Gordon Off the top of my head, I cannot think of a better method; that's why I went with this one ;-) Jun 27, 2018 at 21:15
• Better? Maybe a fit to a Chebyshev series. Details left as an exercise. There are whole books on the art of function approximation. Jun 27, 2018 at 21:18
• @AccidentalFourierTransform thanks. Out of curiousity, where would you divide by (\Delta x), as I need the actual values of the derivative, not just the shape. Jun 27, 2018 at 21:36