interpolation - 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:

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{2.19`, 1.18058665164381804732895291445055512447`10.150319820029056}, \
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{2.22`, 1.18053066355023167469223980968670030235`10.150319904811823}, \

{2.23`, 1.18051433663530613279316577665188132458`10.150319943291501}, \
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{2.2600000000000002`, 
    1.18047358256870323144781205364905150799`10.1503200024423}, \
{2.27`, 1.18045284425836003610183819602310982111`10.15032006948038}, \
{2.2800000000000002`, 
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{2.29`, 1.18042205490595340640316049868951433544`10.150320139600492}, \
{2.3000000000000003`, 

    1.1804080664330128916841858733953962064`10.150320174993}, {2.31`, 
    1.18038941529283831613907223168965142054`10.150320177542772}, \
{2.32`, 1.18038455410165951359586395404854568313`10.150320276480821}, \
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{2.39`, 1.18029315613612538342372210272335426953`10.150320616336812}, \

{2.4`, 1.18027136016388886609024108419546052058`10.15032049953037}, \
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{2.46`, 1.18019936285550823302632808088898016066`10.150320637244697}, \
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{2.49`, 1.18016459448109308818327022400910753323`10.150320604856626}, \

{2.5`, 1.18015519313440986415250313960183004747`10.15032074122922}, \
{2.5100000000000002`, 
    1.1801395093394907109725119477046775641`10.150320673420158}, \
{2.52`, 1.1801521189103215413303663227645363953`10.150320879595276}, \
{2.5300000000000002`, 
    1.1801307972602899695356580794903075229`10.150320703866537}, \
{2.54`, 1.1801219032572064046637548898437084353`10.15032077822856}, \
{2.5500000000000003`, 
    1.18010778328630934425504963985643981838`10.150320780679827}, \
{2.56`, 1.1800981476132935685234927964634260708`10.150320797194558}, \

{2.57`, 1.1800852880880741598233297715693878691`10.150320812448818}, \
{2.58`, 1.1800803367361048957098679244058279423`10.150320835462551}, \
{2.59`, 1.18007131998070713493031563275406521632`10.150320850568097}, \
{2.6`, 1.18006209240679324087354930157845795277`10.150320869927336}, \
{2.61`, 1.18005388978067128110124165523443609432`10.150320887923487}, \
{2.62`, 1.18004507530208995556184542018796405579`10.150320901991472}, \
{2.63`, 1.18003682106836963817060152637066436015`10.150320917134115}, \
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{2.66`, 1.18001302942814922306819489909187068756`10.150320969148144}, \

{2.67`, 1.18000585845323368595610623797510024537`10.150320986058905}, \
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{2.75`, 1.17995230170237949651812671147579581646`10.15032108285379}, \
{2.7600000000000002`, 

    1.17994612843326319735757104739063728625`10.150321099861214}, \
{2.77`, 1.17993882505107265878314852070597855445`10.150321086933197}, \
{2.7800000000000002`, 
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{2.79`, 1.17992909870801670694847287322627336658`10.150321131596627}, \
{2.8000000000000003`, 
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{2.81`, 1.17991912467737054169571074322045806068`10.15032114225105}, \
{2.82`, 1.17991480940324422553226187393945756974`10.150321154100451}, \
{2.83`, 1.17990958874666267274774447524562975543`10.150321181093055}, \

{2.84`, 1.17990564194203696094776943735162938735`10.150321172747566}, \
{2.85`, 1.17989994927618164916452938552678725266`10.15032120730178}, \
{2.86`, 1.17989755501309123951355728961865913188`10.150321198361539}, \
{2.87`, 1.17989232754353003171510898983896799535`10.150321194894682}, \
{2.88`, 1.17988845568961326992564992642119012053`10.150321197127615}, \
{2.89`, 1.17988589742719803584973730650738260995`10.150321212434863}, \
{2.9`, 1.17988176581893780962421803300552151788`10.15032119417042}, \
{2.91`, 1.17987875417937959585613984639853561569`10.150321197195288}, \
{2.92`, 1.17987965938957754449600847664661010376`10.150321205153638}, \
{2.93`, 1.17987402613409458459515189806653082172`10.150321201971883}, \

{2.94`, 1.17987222533192294311177015368629973929`10.150321206144982}, \
{2.95`, 1.17986957347002501590911694223837354359`10.150321212486556}, \
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{2.97`, 1.17986323710674197821728392192023793707`10.150321215599114}, \
{2.98`, 1.17985949640129442747227095859703509117`10.150321207294247}, \
{2.99`, 1.17985551713664815217415823611544887084`10.15032118932838}, \
{3.`, 1.17985149480779907471182091433276932591`10.150321160381248}, \
{3.0100000000000002`, 
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{3.02`, 1.17984424326886577914136687162659965423`10.150321065700378}, \

{3.0300000000000002`, 
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{3.04`, 1.17983972280714366167920448006408726134`10.150320931432793}, \
{3.0500000000000003`, 
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{3.06`, 1.17983805672222088042509478174748495443`10.150320777479246}, \
{3.0700000000000003`, 
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{3.08`, 1.17984010278916593097573438815866905194`10.15032062546959}, \
{3.09`, 1.17984090117462542124244791687790076279`10.150320559809153}, 

   {3.1`, 
    1.17982997239924707255835635158905814981`10.150320733070526}, \
{3.11`, 1.17984393058236338402998580037742437307`10.150320451743152}, \
{3.12`, 1.1798458854886468657719264206187240623`10.150320429716823}, \
{3.13`, 1.17984624533975615219754358686757348567`10.150320394862455}, \
{3.14`, 1.17984664133333547726439513516606186835`10.150320383670744}};

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