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Derivatae fvnctionvm elementarivm

Svmmarivm

Haec lectio fvnctiones elementares derivat qvas lectio svperior non attigit: sinvm et cosinvm, ex limitibvs trigonometricis partis 1; tangentem, cotangentem, secantem et cosecantem, regvlis qvotientis et inversi; exponentialem natvralem et logarithmvm natvralem, ex limitibvs eorvm notabilibvs; fvnctiones deniqve hyperbolicas, ex exponentiali. Nvlla demonstratio regvla catenae avt regvla fvnctionis inversae vtitvr, qvae postea venivnt: qvoqve casv formvla additionis avt lex exponentialis vel logarithmi qvotientem incrementalem in factorem dividit qvi ex solo pvncto pendet et in altervm qvi limes iam notvs est. His derivatis et regvlis lectio rectas tangentes et angvlos inter cvrvas transcendentes compvtat, et qvod lectio svperior promisit praestat: fvnctio x2sin⁡1xx^2 \sin \frac{1}{x}, in 00 derivabilis, derivatam sine limite in 00 habet.

Proposita discendi

  1. Demonstrare derivatam sinvs esse cosinvm et derivatam cosinvs esse oppositvm sinvs, et explicare cvr variabilis in radianis metiatvr.
  2. Tangentem, cotangentem, secantem et cosecantem regvlis lectionis svperioris derivare, et dominivm cvivsqve derivatae definire.
  3. Demonstrare exponentialem natvralem svam ipsivs derivatam esse et derivatam fvnctionis ln⁡∣x∣\ln \abs{x} esse 1x\frac{1}{x}, atqve inde derivatas fvnctionvm hyperbolicarvm obtinere.
  4. Derivatas elementares cvm regvlis conivngere vt rectae tangentes et angvli inter cvrvas inveniantvr, et agnoscere derivatam posse limitem non habere in pvncto vbi fvnctio derivabilis est.

Sinvs et cosinvs

Lectio de regvlis derivandi omnes fvnctiones derivavit qvae ex identitate et constantibvs per qvattvor operationes obtinentvr: polynomia et fvnctiones rationales. Sinvs non est ex eis, et derivata eivs ex qvotiente incrementali svo oriri debet. In 00 hic qvotiens est sin⁡hh\frac{\sin h}{h}, et limes fvndamentalis trigonometriae, ex lectione de theoremate compressionis, derivatam 11 dat. Pro pvncto aa qvolibet, formvla sinvs svmmae sin⁡(a+h)\sin(a + h) in terminvm dividit qvi ex solo aa pendet et in altervm qvi ex solo hh pendet; qvotientes qvi restant dvo limites svnt corollarii de limitibvs trigonometricis dedvctis, ex eadem lectione: sin⁡hh\frac{\sin h}{h} ad 11 tendit, et 1−cos⁡hh\frac{1 - \cos h}{h} ad 00.

TheoremaDerivatae sinvs et cosinvs

Sinvs et cosinvs in toto R\mathbb{R} derivabiles svnt, et pro omni reali xx:

ddx(sin⁡x)=cos⁡x,ddx(cos⁡x)=−sin⁡x\frac{\mathrm{d}}{\mathrm{d}x}\left(\sin x\right) = \cos x, \qquad \frac{\mathrm{d}}{\mathrm{d}x}\left(\cos x\right) = -\sin x

Adhibentvr forma incrementalis derivatae, ex lectione de derivata vt limite, et formvlae additionis ex lectione de formvlis additionis et svbtractionis. Figvra rectam tangentem delineationis sinvs comitatvr: declivitas eivs in qvoqve pvncto est ordinata cosinvs in eodem pvncto; in fine, recta tangens delineationis cosinvs declivitatem habet oppositvm sinvs.

Demonstratio

  1. sin⁡(a+h)−sin⁡a=sin⁡acos⁡h+cos⁡asin⁡h−sin⁡a=sin⁡a(cos⁡h−1)+cos⁡asin⁡h\sin(a + h) - \sin a = \sin a \cos h + \cos a \sin h - \sin a = \dato{s}{\sin a \left(\cos h - 1\right) + \cos a \sin h}

    Sint aa nvmervs realis et h≠0h \neq 0. Per formvlam sinvs svmmae, incrementvm sinvs in dvos terminos dividitvr: altervm cvm factore cos⁡h−1\cos h - 1, altervm cvm factore sin⁡h\sin h; factores sin⁡a\sin a et cos⁡a\cos a ex hh non pendent.

  2. sin⁡(a+h)−sin⁡ah=−sin⁡a⋅1−cos⁡hh+cos⁡a⋅sin⁡hh→−sin⁡a⋅0+cos⁡a⋅1=cos⁡a\frac{\sin(a + h) - \sin a}{h} = -\sin a \cdot \frac{1 - \cos h}{h} + \cos a \cdot \frac{\sin h}{h} \to -\sin a \cdot 0 + \cos a \cdot 1 = \resaltar{\cos a}

    Dividitvr per hh. Per corollarivm de limitibvs trigonometricis dedvctis, dvo qvotientes ad 00 et ad 11 tendvnt; per theorema de limite combinationis linearis, ex algebra limitvm, qvotiens incrementalis limitem habet, et forma incrementalis derivatam sinvs in aa dat.

  3. cos⁡(a+h)−cos⁡ah=−cos⁡a⋅1−cos⁡hh−sin⁡a⋅sin⁡hh→−cos⁡a⋅0−sin⁡a⋅1=−sin⁡a\frac{\cos(a + h) - \cos a}{h} = -\cos a \cdot \frac{1 - \cos h}{h} - \sin a \cdot \frac{\sin h}{h} \to -\cos a \cdot 0 - \sin a \cdot 1 = \resaltar{-\sin a}

    Formvla cosinvs svmmae, cos⁡(a+h)=cos⁡acos⁡h−sin⁡asin⁡h\cos(a + h) = \cos a \cos h - \sin a \sin h, idem ratiocinivm derivatam cosinvs dat.

Demonstratio vtitvr eo qvod variabilis in radianis metitvr: limes fvndamentalis sinvm cvm longitvdine arcvs comparat. Si angvlvs in gradibvs metiretvr, sinvs xx gradvvm esset sin⁡πx180\sin \frac{\pi x}{180}, et derivata eivs factorem π180\frac{\pi}{180} ferret (problema propositvm 1). Radiani igitvr svnt mensvra in qva sinvs derivatam habet cosinvm, sine factoribvs. Interpretatio geometrica theorematis est ea figvrae: sinvs crescit vbi cosinvs positivvs est et rectam tangentem horizontalem habet vbi cosinvs evanescit; recta tangens sinvs in origine est recta y=xy = x, et ideo prope 00 valor sin⁡x\sin x similis est valori xx, vt iam aeqvivalentia sin⁡x∼x\sin x \sim x lectionis de infinitesimis aeqvivalentibvs dicebat.