Derivatae: synthesis et provocationes

Svmmarivm

Haec lectio partem de derivatis clavdit. In vnam catenam dedvctivam colligit qvae decem lectiones partis demonstravervnt: derivatam vt limitem qvotientis incrementalis, tres eivs characterismos aeqvivalentes et regvlas derivandi, omnia ex factorizatione Carathéodorii obtenta sine divisione per qvantitates qvae evanescere possvnt, vna cvm derivatis fvnctionvm elementarivm et regvla catenae; theorema valoris intermedii et continvitatem inversae, qvae derivatam fvnctionis inversae svstinent, et redvctionem potentiarvm et exponentialivm cvivsvis basis ad exponentialem natvralem; et derivatas svccessivas, differentiale et rationes mvtationis. Explicat qvomodo pars parti svperiori et Algebrae et Geometriae I et II innitatvr, et qvomodo applicationes derivatarvm praeparet. Desinit in probatione finali, initiali parallela, et in decem provocationibvs, difficilioribvs qvam problemata lectionvm, qvarvm vnaqvaeqve effectvs plvrivm earvm conivngit.

Proposita discendi

Definitiones et theoremata praecipva partis ordine logico exponere, indicando ex qvibvs effectibvs svperioribvs qvodqve pendeat et in qvibvs demonstrationibvs factorizatio Carathéodorii itervm adhibeatvr.
Distingvere qvae pars admittat ab iis qvae demonstret, et hypotheses cvivsqve effectvs agnoscere, vt eam derivatae inversae vel eam qvam derivatio implicita postvlat.
Effectvs partis cvm parte de limitibvs et de continvitate, cvm Algebra et Geometria I et II et cvm applicationibvs derivatarvm conferre.
Problemata non vsitata solvere qvae definitionem derivatae, regvlas, regvlam catenae, theorema valoris intermedii, derivatam inversae, potentias exponentis variabilis, derivatas svccessivas, differentiale et rationes mvtationis conivngvnt.

Qvae vnitas consecvta est

Tota pars vni ideae innititvr: derivata est limes, et pars svperior iam omnia praebverat qvae ad eam compvtandam et tractandam reqvirvntvr. Sed qvotiens incrementalis est divisio, et per incrementvm qvod evanescere potest dividere origo est demonstrationvm vitiosarvm. Characterismvs Carathéodorii divisioni prodvctvm svbstitvit, cvm factore $\varphi$ in $a$ continvo:

f(x) - f(a) = \varphi(x)(x - a), \qquad \varphi(a) = f'(a)

Eo adhibito, demonstrare fvnctionem derivabilem esse eo redigitvr vt eivs incrementvm factore continvo in factores resolvatvr, et de continvitate factoris theoremata partis svperioris decernvnt. Sic obtentae svnt regvlae combinationis linearis, prodvcti et inversi, regvla catenae et derivata fvnctionis inversae, qvaeqve svperiores citans loco earvm itervm faciendarvm.

Definitio et regvlae

Lectio de derivata vt limite ad eandem definitionem a recta tangente et a velocitate pervenit, et statvit vbi sensvm habeat: in pvncto dominii qvod eivs pvnctvm accvmvlationis sit. Derivatam vt envntiatvm qvantificatvm scripsit, derivatas laterales vt limites laterales qvotientis incrementalis definivit, rectam tangentem et rectam normalem, fvnctionem derivatam et notationem Leibnitianam, et ostendit derivatam tres formas aeqvivalentes admittere: limitem qvotientis incrementalis, approximationem $f(a + h) = f(a) + f'(a)h + \operatorname{o}(h)$ et factorizationem Carathéodorii. Ex vltima dvobvs versibvs secvtvm est fvnctionem derivabilem continvam esse; valor absolvtvs et radix cvbica ostendervnt propositionem conversam non valere. Lectio de regvlis derivandi primvm demonstravit derivatam tantvm ex eo pendere qvod prope pvnctvm accidit, deinde, factorizationibvs conivnctis, regvlas combinationis linearis, prodvcti, inversi et qvotientis; iis potentias exponentis integri, polynomia et fvnctiones rationales derivavit, et de derivabilitate in pvncto ivnctionis dvorvm frvstorvm decrevit, vbi derivatae formvlarvm non svfficivnt. Lectio de derivatis fvnctionvm elementarivm eas sinvs et cosinvs ex limite trigonometrico fvndamentali et ex formvlis additionis obtinvit, eas exponentialis natvralis et ipsivs $\ln \abs{x}$ ex limitibvs notabilibvs, sine regvla catenae et sine regvla fvnctionis inversae, et eas ceterarvm fvnctionvm trigonometricarvm et fvnctionvm hyperbolicarvm per regvlas. Lectio de regvla catenae dvas factorizationes conivnxit, explicavit cvr demonstratio qvae per $g(x) - g(a)$ dividit deficiat cvm fvnctio interior valorem svvm iterat, dominivm compositionis cvravit et regvlam notatione Leibnitiana scripsit, vbi simplificatio fractionvm videtvr neqve est.

Continvitas, inversae et exponentes

Lectio de theoremate valoris intermedii theorema Bolzani demonstravit per svpremvm pvnctorvm in qvibvs fvnctio negativa est et per conservationem signi, et ex eo theorema valoris intermedii obtinvit, imaginem intervalli per fvnctionem continvam intervallvm esse et inversam fvnctionis continvae et stricte monotonae in intervallo continvam esse; methodvs bisectionis demonstrationem in rationem procedendi convertit. Lectio de derivata fvnctionis inversae factorizationem Carathéodorii retrorsvm legit: si $f'(a) \neq 0$ , factor non evanescit, et eivs inversvm, cvm inversa continva compositvm, incrementvm inversae in factores resolvit; si $f'(a) = 0$ , regvla catenae impedit qvominvs inversa derivabilis sit. Eo theoremate fvnctiones trigonometricas inversas, fvnctiones hyperbolicas inversas, radices et potentias exponentis rationalis derivavit, et fvnctiones implicite definitas derivavit, qvoqve casv svmens eas exsistere et derivabiles esse. Lectio de variabili in exponente omnia ad exponentialem natvralem redegit, per $a^x = e^{x \ln a}$ et $\left(f(x)\right)^{g(x)} = e^{g(x) \ln f(x)}$ : exponentialem et logarithmvm cvivsvis basis, potentias exponentis realis et eas basis et exponentis variabilivm derivavit, derivationem logarithmicam ivstificavit derivabilitate demonstrata anteqvam logarithmi svmerentvr, et decrevit $\pi^e < e^\pi$ per rectam tangentem logarithmi qvae per originem transit.

Derivatae svccessivae et rationes mvtationis

Lectio de derivatis svccessivis et differentiali derivatas svccessivas per recvrsionem definivit, cvm dominio svo, et per indvctionem formvlas ordinis $n$ demonstravit pro potentiis, exponentiali, sinv et cosinv, et regvlam Leibnitianam, qvae coefficientes binomii Newtoniani habet. Ipsis $\mathrm{d}y$ et $\mathrm{d}x$ significationem separatam dedit: differentiale $\mathrm{d}y = f'(x) \, \mathrm{d}x$ est vnica pars linearis incrementi qvae ab eo qvantitate differt qvae respectv ipsivs $\Delta x$ neglegi potest; eo vsa est ad valores approximandos et ad propagationem errorvm aestimandam, nvlla erroris approximationis margine statvta, qvod instrvmenta posteriora postvlat. Lectio de rationibvs mvtationis derivatam legit tamqvam mensvram qva celeritate qvantitas mvtetvr: rationes mediam, momentaneam et relativam distinxit, incrementvm ratione relativa constanti proprie descripsit, motvm rectilinevm per velocitatem, celeritatem et accelerationem descripsit, demonstravit qvid signvm derivatae in pvncto de valoribvs vicinis dicat, nec plvs, et methodvm explicitam ad rationes mvtationis conexas in regvla catenae fvndavit.

Qvae admittvntvr, qvae demonstrantvr, qvae restant

Ratio partis sic svmmatim reddi potest:

Admittvntvr axioma completvdinis, qvo in hac parte solvm theorema Bolzani directe vtitvr, theoremata partis de limitibvs et de continvitate et effectvs in Algebra et Geometria demonstrati: formvlae additionis, identitates fvndamentales, leges exponentivm et logarithmorvm, inaeqvalitas $e^x \geq 1 + x$ , monotonia fvnctionvm qvae inversas definivnt, indvctio et binomivm Newtonianvm. Svmitvr etiam, qvoqve casv, aeqvationem fvnctionem derivabilem definire cvm implicite derivatvr.
Demonstrantvr facta qvae Algebra et Geometria admiserat vel nvntiaverat: theorema valoris intermedii, rectam tangentem delineationis ipsivs $e^x$ in $(0, 1)$ declivitatem $1$ habere et $e$ vnicam basim esse cvivs exponentialis eam declivitatem habeat; et recta tangens, qvae pro conicis definita erat vt recta qvae cvrvam tangit neqve secat, pro qvalibet delineatione definitvr.
Qvattvor instrvmenta itervm adhibentvr loco argvmentorvm repetendorvm: factorizatio Carathéodorii, theorema qvo fvnctio derivabilis continva est, indoles localis derivatae et conservatio signi.
In postervm differtvr qvod plvs qvam derivatam in pvncto postvlat: theorema valoris medii et qvae ex eo colligvntvr, monotonia ex signo derivatae, extrema, regvla Hospitalii et margines erroris differentialis, in parte proxima; polynomivm Taylorianvm et integrale, in Calcvlo II; et theorema qvod fvnctiones implicite definitas praestat, in calcvlo plvrivm variabilivm.

Pictvra qvae svbicitvr tria segmenta in vnam catenam colligit. Qvisqve nodvs colvmnae est definitio avt theorema, secvndvm ordinem logicvm nvmeratvs et arcv cvm anvlis svperioribvs conivnctvs qvibvs eivs demonstratio vtitvr; circvm flvitant fvlcra, id est principia et instrvmenta ad qvae catena itervm atqve itervm recvrrit, qvodqve linea continva cvm anvlis qvi ei nitvntvr conivnctvm et linea pvnctata cvm anvlo vbi statvitvr, exceptis iis qvae ex parte svperiore vel ex Algebra et Geometria provenivnt, qvae vt circvli vacvi pingvntvr. Fvlcro pvlsato omnes anvli qvi eo vtvntvr illvstrantvr, et qvisqve anvlvs ad lectionem remittit in qva demonstratvr.

Fvlcra catenae

AAxioma completvdinis. Ex Algebra et Geometria I admittitvr reales corpvs ordinatvm completvm efficere: qvaevis copia realivm non vacva svperne circvmscripta svpremvm habet. In hac vnitate sine demonstratione admittitvr.
BHereditas vnitatis 1. Effectvs in parte de limitibvs et de continvitate demonstrati: algebra limitvm, cvm limite restrictionis et eo compositionis; limites laterales; continvitas, ea operationvm et ea compositionis; continvitas fvnctionvm elementarivm et inversae fvnctionis monotonae cvivs imago intervallvm est; limites ipsorvm $\frac{\sin h}{h}$ , $\frac{1 - \cos h}{h}$ , $\frac{e^h - 1}{h}$ et $\frac{\ln(1 + h)}{h}$ in $0$ ; et $\operatorname{o}$ parvvm et aeqvivalentia. In hac vnitate sine demonstratione admittitvr.
CConservatio signi. In parte de limitibvs et de continvitate demonstrata: si $f$ in $a$ ad limitem $L \neq 0$ tendit, prope $a$ fvnctio signvm ipsivs $L$ habet; et si $f$ in $a$ continva est cvm $f(a) \neq 0$ , prope $a$ signvm ipsivs $f(a)$ habet neqve evanescit. In hac vnitate sine demonstratione admittitvr.
DHereditas Algebrae et Geometriae. Effectvs in Algebra et Geometria I et II demonstrati: indvctio, binomivm Newtonianvm et regvla Pascalis; formvlae additionis et identitates fvndamentales trigonometricae et hyperbolicae; leges exponentivm et logarithmorvm, mvtatio basis et inaeqvalitas $e^x \geq 1 + x$ ; et monotonia restrictionvm qvae fvnctiones trigonometricas inversas definivnt et fvnctionvm hyperbolicarvm. In hac vnitate sine demonstratione admittitvr.
EFactorizatio Carathéodorii. $f$ in $a$ derivabilis est, cvm $f'(a) = L$ , si et solvm si exsistit $\varphi$ , in $a$ continva, cvm $\varphi(a) = L$ et $f(x) - f(a) = \varphi(x)(x - a)$ in toto dominio: ad derivandvm svfficit incrementvm in factores resolvere, sine divisione. Constitvitvr in: 3. Tres characterismi.
FDerivabilis ergo continva. Omnis fvnctio in pvncto derivabilis ibi continva est; regvlae hoc adhibent vt factor Carathéodorii continvvs evadat. Constitvitvr in: 4. Derivabilis ergo continva.
GIndoles localis. Derivata in $a$ non mvtatvr cvm fvnctio ad copiam restringitvr cvivs $a$ adhvc pvnctvm accvmvlationis est, et tantvm ex valoribvs prope $a$ pendet; ideo in intersectione dominiorvm vel in intervallo aperto laboratvr. Constitvitvr in: 5. Indoles localis derivatae.

Segmentvm 1 Definitio et regvlae
- Derivatam in pvncto accvmvlationis dominii vt limitem qvotientis incrementalis definire, cvm derivatis eivs lateralibvs, et eam vt envntiatvm qvantificatvm scribere.
- Aeqvivalentiam qvotientis incrementalis, approximationis linearis et factorizationis Carathéodorii demonstrare, et inde dedvcere fvnctionem derivabilem continvam esse, non vicissim.
- Factorizationibvs, sine divisione per incrementa, regvlas combinationis linearis, prodvcti, inversi et qvotientis obtinere, et polynomia, fvnctiones rationales et fvnctiones per frvsta definitas derivare.
- Fvnctiones trigonometricas, exponentialem, logarithmvm et fvnctiones hyperbolicas per limites partis svperioris derivare, et earvm compositiones per regvlam catenae.
1. Definitio derivatae. Si $a$ est elementvm dominii $A$ fvnctionis $f$ qvod pvnctvm accvmvlationis ipsivs $A$ est, tvm $f'(a) = L$ cvm $(\forall \varepsilon > 0)(\exists \delta > 0)(\forall x \in A)\left(0 < \abs{x - a} < \delta \rightarrow \abs{\frac{f(x) - f(a)}{x - a} - L} < \varepsilon\right)$ ; aeqvivalenter, $\lim_{h \to 0} \frac{f(a + h) - f(a)}{h} = L$ .
  Nititvr: B (Hereditas vnitatis 1). Demonstratvr in: Derivata vt limes fvnctionis.
2. Derivatae laterales. Derivatae laterales $f'_+(a)$ et $f'_-(a)$ svnt limites laterales qvotientis incrementalis; si $a$ est pvnctvm accvmvlationis dominii ab vtroqve latere, $f$ in $a$ derivabilis est si et solvm si ambae exsistvnt et aeqvales svnt, et si ab vno tantvm latere est, derivata est ea illivs lateris.
  Vtitvr: 1. Definitio derivatae. Nititvr: B (Hereditas vnitatis 1). Demonstratvr in: Derivata vt limes fvnctionis.
3. Tres characterismi. Aeqvivalent: $f'(a) = L$ ; $f(a + h) = f(a) + Lh + \operatorname{o}(h)$ cvm $h$ ad $0$ tendit; et exsistentia fvnctionis $\varphi$ , in $a$ continvae, cvm $\varphi(a) = L$ , ita vt $f(x) - f(a) = \varphi(x)(x - a)$ pro omni $x$ dominii.
  Vtitvr: 1. Definitio derivatae. Nititvr: B (Hereditas vnitatis 1). Demonstratvr in: Derivata vt limes fvnctionis.
4. Derivabilis ergo continva. Si $f$ in $a$ derivabilis est, in $a$ continva est; propositio conversa falsa est: $\abs{x}$ et $\sqrt[3]{x}$ in $0$ continvae svnt et in $0$ derivabiles non svnt.
  Vtitvr: 2. Derivatae laterales, 3. Tres characterismi. Nititvr: B (Hereditas vnitatis 1), E (Factorizatio Carathéodorii). Demonstratvr in: Derivata vt limes fvnctionis.
5. Indoles localis derivatae. Si $F$ est restrictio fvnctionis $f$ ad copiam $B$ qvae $a$ continet et cvivs $a$ pvnctvm accvmvlationis est, et $f$ in $a$ derivabilis est, tvm $F'(a) = f'(a)$ ; et si $B$ omnia pvncta dominii prope $a$ continet, $f$ in $a$ derivabilis est si et solvm si $F$ est.
  Vtitvr: 1. Definitio derivatae. Nititvr: B (Hereditas vnitatis 1). Demonstratvr in: Regvlae derivandi.
6. Combinatio linearis et prodvctvm. Si $f$ et $g$ in $a$ derivabiles svnt, etiam $\alpha f + g$ et $fg$ svnt, cvm $(\alpha f + g)'(a) = \alpha f'(a) + g'(a)$ et $(fg)'(a) = f'(a) \, g(a) + f(a) \, g'(a)$ .
  Vtitvr: 3. Tres characterismi, 4. Derivabilis ergo continva. Nititvr: B (Hereditas vnitatis 1), E (Factorizatio Carathéodorii), F (Derivabilis ergo continva). Demonstratvr in: Regvlae derivandi.
7. Inversvm et qvotiens. Si $f$ et $g$ in $a$ derivabiles svnt et $g(a) \neq 0$ , tvm $a$ pvnctvm accvmvlationis dominii ipsivs $\frac{f}{g}$ est, et valent $\left(\frac{1}{g}\right)'(a) = -\frac{g'(a)}{\left(g(a)\right)^2}$ et $\left(\frac{f}{g}\right)'(a) = \frac{f'(a) \, g(a) - f(a) \, g'(a)}{\left(g(a)\right)^2}$ .
  Vtitvr: 3. Tres characterismi, 4. Derivabilis ergo continva, 5. Indoles localis derivatae, 6. Combinatio linearis et prodvctvm. Nititvr: B (Hereditas vnitatis 1), C (Conservatio signi), E (Factorizatio Carathéodorii), F (Derivabilis ergo continva), G (Indoles localis). Demonstratvr in: Regvlae derivandi.
8. Potentiae, polynomia et fvnctiones rationales. Pro omni integro $m \neq 0$ , derivata ipsivs $x^m$ est $m x^{m - 1}$ in toto eivs dominio; omne polynomivm in $\mathbb{R}$ derivabile est, cvm derivata termino post terminvm, et omnis fvnctio rationalis in svo dominio derivabilis est.
  Vtitvr: 1. Definitio derivatae, 6. Combinatio linearis et prodvctvm, 7. Inversvm et qvotiens. Nititvr: D (Hereditas Algebrae et Geometriae). Demonstratvr in: Regvlae derivandi.
9. Derivata in pvncto ivnctionis. Si $f$ cvm $g$ a sinistra ipsivs $c$ congrvit et cvm $h$ a dextra, cvm $g$ et $h$ in $c$ derivabilibvs, tvm $f$ in $c$ derivabilis est si et solvm si $g(c) = f(c) = h(c)$ et $g'(c) = h'(c)$ ; derivatas formvlarvm comparare non svfficit.
  Vtitvr: 2. Derivatae laterales, 4. Derivabilis ergo continva. Nititvr: B (Hereditas vnitatis 1), F (Derivabilis ergo continva). Demonstratvr in: Regvlae derivandi.
10. Fvnctiones trigonometricae. Pro omni $x$ reali, $\frac{\mathrm{d}}{\mathrm{d}x}\left(\sin x\right) = \cos x$ et $\frac{\mathrm{d}}{\mathrm{d}x}\left(\cos x\right) = -\sin x$ ; vbi $\cos x \neq 0$ , $\frac{\mathrm{d}}{\mathrm{d}x}\left(\tan x\right) = \frac{1}{\cos^2 x} = 1 + \tan^2 x$ ; et cotangens, secans et cosecans per regvlas qvotientis et inversi derivantvr.
  Vtitvr: 1. Definitio derivatae, 6. Combinatio linearis et prodvctvm, 7. Inversvm et qvotiens. Nititvr: B (Hereditas vnitatis 1), D (Hereditas Algebrae et Geometriae). Demonstratvr in: Derivatae fvnctionvm elementarivm.
11. Exponentialis, logarithmvs et hyperbolicae. $\frac{\mathrm{d}}{\mathrm{d}x}\left(e^x\right) = e^x$ in $\mathbb{R}$ , $\frac{\mathrm{d}}{\mathrm{d}x}\left(\ln \abs{x}\right) = \frac{1}{x}$ in $\mathbb{R} \setminus \{0\}$ , et derivatae ipsarvm $\sinh x$ , $\cosh x$ et $\tanh x$ svnt $\cosh x$ , $\sinh x$ et $\frac{1}{\cosh^2 x}$ .
  Vtitvr: 1. Definitio derivatae, 6. Combinatio linearis et prodvctvm, 7. Inversvm et qvotiens. Nititvr: B (Hereditas vnitatis 1), D (Hereditas Algebrae et Geometriae). Demonstratvr in: Derivatae fvnctionvm elementarivm.
12. Regvla catenae. Si $g$ in $a$ derivabilis est et $f$ , in copia definita qvae valores ipsivs $g$ continet, in $g(a)$ derivabilis est, tvm $(f \circ g)'(a) = f'(g(a)) \, g'(a)$ ; demonstratio qvae per $g(x) - g(a)$ dividit valet tantvm cvm $g'(a) \neq 0$ .
  Vtitvr: 3. Tres characterismi, 4. Derivabilis ergo continva. Nititvr: B (Hereditas vnitatis 1), C (Conservatio signi), E (Factorizatio Carathéodorii), F (Derivabilis ergo continva). Demonstratvr in: Regvla catenae ad derivandam compositionem fvnctionvm.
13. Compositiones vsitatae. Regvla in dominio ipsivs $f \circ g$ valet cvm $a$ eivs pvnctvm accvmvlationis est, qvod accidit si dominivm ipsivs $f$ intervallvm centrvm $g(a)$ habens continet; ea adhibita, $\frac{\mathrm{d}}{\mathrm{d}x}\left(\left(g(x)\right)^m\right) = m \left(g(x)\right)^{m - 1} g'(x)$ pro omni integro $m \neq 0$ , $\frac{\mathrm{d}}{\mathrm{d}x}\left(\sqrt{g(x)}\right) = \frac{g'(x)}{2\sqrt{g(x)}}$ si $g(a) > 0$ , et $\frac{\mathrm{d}}{\mathrm{d}x}\left(\sin g(x)\right) = \cos g(x) \cdot g'(x)$ .
  Vtitvr: 1. Definitio derivatae, 5. Indoles localis derivatae, 8. Potentiae, polynomia et fvnctiones rationales, 10. Fvnctiones trigonometricae, 12. Regvla catenae. Nititvr: C (Conservatio signi), F (Derivabilis ergo continva), G (Indoles localis). Demonstratvr in: Regvla catenae ad derivandam compositionem fvnctionvm.
Segmentvm 2 Continvitas, inversae et exponentes
- Theorema Bolzani per svpremvm demonstrare et, eo adhibito, theorema valoris intermedii, imaginem intervalli per fvnctionem continvam intervallvm esse et inversam fvnctionis continvae et stricte monotonae continvam esse.
- Derivatam fvnctionis inversae demonstrare factorizatione Carathéodorii retrorsvm lecta, et fvnctiones trigonometricas inversas, fvnctiones hyperbolicas inversas, radices et potentias exponentis rationalis derivare.
- Fvnctiones implicite definitas derivare, compvtationem derivatae a demonstratione eam exsistere distingvendo.
- Ad exponentialem natvralem redigere exponentiales et logarithmos cvivsvis basis et potentias exponentis realis vel basis et exponentis variabilivm, et derivationem logarithmicam ivstificare.
1. Theorema Bolzani. Si $f$ in $\icc{a}{b}$ continva est et $f(a) < 0 < f(b)$ , vel $f(a) > 0 > f(b)$ , exsistit $c$ in $\ioo{a}{b}$ ita vt $f(c) = 0$ .
  Nititvr: A (Axioma completvdinis), C (Conservatio signi). Demonstratvr in: Theorema valoris intermedii.
2. Theorema valoris intermedii. Si $f$ in $\icc{a}{b}$ continva est, omnem valorem inter $f(a)$ et $f(b)$ assvmit; ideo imago intervalli per fvnctionem continvam intervallvm est.
  Vtitvr: 14. Theorema Bolzani. Nititvr: B (Hereditas vnitatis 1). Demonstratvr in: Theorema valoris intermedii.
3. Continvitas inversae. Si $f$ in intervallo $I$ continva et stricte monotona est, eivs inversa in intervallo $f(I)$ continva est.
  Vtitvr: 15. Theorema valoris intermedii. Nititvr: B (Hereditas vnitatis 1). Demonstratvr in: Theorema valoris intermedii.
4. Derivata fvnctionis inversae. Si $f$ in intervallo $I$ plvs qvam vnvm pvnctvm habente continva et stricte monotona est et in $a$ derivabilis, cvm $f'(a) \neq 0$ , tvm $f^{-1}$ in $b = f(a)$ derivabilis est, et $\left(f^{-1}\right)'(b) = \frac{1}{f'(a)}$ ; si $f'(a) = 0$ , $f^{-1}$ in $b$ derivabilis non est.
  Vtitvr: 1. Definitio derivatae, 3. Tres characterismi, 12. Regvla catenae, 15. Theorema valoris intermedii, 16. Continvitas inversae. Nititvr: B (Hereditas vnitatis 1), E (Factorizatio Carathéodorii). Demonstratvr in: Derivata fvnctionis inversae et derivatio implicita.
5. Inversae elementares et radices. In dominiis derivabilitatis svis, $\frac{\mathrm{d}}{\mathrm{d}x}\left(\arcsin x\right) = \frac{1}{\sqrt{1 - x^2}}$ , $\frac{\mathrm{d}}{\mathrm{d}x}\left(\arctan x\right) = \frac{1}{1 + x^2}$ et $\frac{\mathrm{d}}{\mathrm{d}x}\left(\operatorname{arcsinh} x\right) = \frac{1}{\sqrt{x^2 + 1}}$ ; $\sqrt[n]{x}$ in omni $x \neq 0$ svi dominii derivabilis est, non in $0$ ; et $\frac{\mathrm{d}}{\mathrm{d}x}\left(x^{\frac{p}{q}}\right) = \frac{p}{q} x^{\frac{p}{q} - 1}$ in $\ioo{0}{\infty}$ .
  Vtitvr: 8. Potentiae, polynomia et fvnctiones rationales, 10. Fvnctiones trigonometricae, 11. Exponentialis, logarithmvs et hyperbolicae, 13. Compositiones vsitatae, 17. Derivata fvnctionis inversae. Nititvr: B (Hereditas vnitatis 1), D (Hereditas Algebrae et Geometriae). Demonstratvr in: Derivata fvnctionis inversae et derivatio implicita.
6. Exponentialis et logarithmvs basis a. Si $a > 0$ et $a \neq 1$ , $\frac{\mathrm{d}}{\mathrm{d}x}\left(a^x\right) = a^x \ln a$ et $\frac{\mathrm{d}}{\mathrm{d}x}\left(\log_a \abs{x}\right) = \frac{1}{x \ln a}$ ; et si $u$ in pvncto derivabilis est, etiam $e^{u(x)}$ est, cvm derivata $e^{u(x)} u'(x)$ , et si praeterea $u$ ibi non evanescit, etiam $\ln \abs{u(x)}$ est, cvm derivata $\frac{u'(x)}{u(x)}$ .
  Vtitvr: 6. Combinatio linearis et prodvctvm, 11. Exponentialis, logarithmvs et hyperbolicae, 12. Regvla catenae, 13. Compositiones vsitatae. Nititvr: D (Hereditas Algebrae et Geometriae). Demonstratvr in: Variabilis in exponente et derivatio logarithmica.
7. Potentiae exponentis realis et variabilis. Pro omni $r$ reali, $\frac{\mathrm{d}}{\mathrm{d}x}\left(x^r\right) = r x^{r - 1}$ in $\ioo{0}{\infty}$ ; et si $f$ , valoribvs positivis, et $g$ in $a$ derivabiles svnt, tvm $F(x) = \left(f(x)\right)^{g(x)} = e^{g(x) \ln f(x)}$ in $a$ derivabilis est, cvm $F'(a) = \left(f(a)\right)^{g(a)} \left(g'(a) \ln f(a) + \frac{g(a) f'(a)}{f(a)}\right)$ .
  Vtitvr: 6. Combinatio linearis et prodvctvm, 11. Exponentialis, logarithmvs et hyperbolicae, 12. Regvla catenae, 19. Exponentialis et logarithmvs basis a. Nititvr: D (Hereditas Algebrae et Geometriae). Demonstratvr in: Variabilis in exponente et derivatio logarithmica.
8. Derivatio logarithmica. Si $f_1, \ldots, f_k$ in $a$ derivabiles svnt neqve in $a$ evanescvnt, et $f_j$ positiva est cvm $r_j$ integer non est, tvm $F = f_1^{r_1} \cdots f_k^{r_k}$ in $a$ derivabilis est, et $\frac{F'(a)}{F(a)} = r_1 \frac{f_1'(a)}{f_1(a)} + \cdots + r_k \frac{f_k'(a)}{f_k(a)}$ ; derivabilitas demonstratvr anteqvam logarithmi svmantvr.
  Vtitvr: 4. Derivabilis ergo continva, 5. Indoles localis derivatae, 6. Combinatio linearis et prodvctvm, 13. Compositiones vsitatae, 19. Exponentialis et logarithmvs basis a, 20. Potentiae exponentis realis et variabilis. Nititvr: D (Hereditas Algebrae et Geometriae), F (Derivabilis ergo continva), G (Indoles localis). Demonstratvr in: Variabilis in exponente et derivatio logarithmica.
9. Pi elatvm ad e et e elatvm ad pi. Pro omni $x > 0$ valet $x^{\frac{1}{x}} \leq e^{\frac{1}{e}}$ , cvm aeqvalitate tantvm si $x = e$ , qvia logarithmvs infra rectam tangentem svam $y = \frac{x}{e}$ iacet; praesertim $\pi^e < e^\pi$ .
  Vtitvr: 11. Exponentialis, logarithmvs et hyperbolicae. Nititvr: D (Hereditas Algebrae et Geometriae). Demonstratvr in: Variabilis in exponente et derivatio logarithmica.
Segmentvm 3 Derivatae svccessivae et rationes mvtationis
- Derivatas svccessivas definire et per indvctionem formvlas ordinis $n$ pro potentiis, exponentiali, sinv et cosinv, et regvlam Leibnitianam demonstrare.
- Ipsis $\mathrm{d}y$ et $\mathrm{d}x$ significationem separatam dare, demonstrare differentiale esse partem principalem incrementi eoqve vti ad approximandvm et ad errores aestimandos, nvlla eorvm margine statvta.
- Derivatam vt rationem mvtationis mediam, momentaneam et relativam legere, eaqve motvm rectilinevm describere.
- Problemata rationvm mvtationis conexarvm per regvlam catenae et methodvm explicitam solvere.
1. Derivatae svccessivae. $f^{(0)} = f$ et $f^{(n + 1)} = \left(f^{(n)}\right)'$ , cvm dominio cvivsqve ex pvnctis constante in qvibvs derivata antecedens derivabilis est; si $f$ et $g$ in intervallo aperto $n$ vicibvs derivabiles svnt, $\left(\alpha f + g\right)^{(n)} = \alpha f^{(n)} + g^{(n)}$ .
  Vtitvr: 6. Combinatio linearis et prodvctvm. Nititvr: D (Hereditas Algebrae et Geometriae), G (Indoles localis). Demonstratvr in: Derivatae svccessivae et differentiale.
2. Derivatae ordinis n. In $\ioo{0}{\infty}$ , $\frac{\mathrm{d}^n}{\mathrm{d}x^n}\left(x^r\right) = r(r - 1)\cdots(r - n + 1) \, x^{r - n}$ ; pro omni $x$ reali, $\frac{\mathrm{d}^n}{\mathrm{d}x^n}\left(e^{cx}\right) = c^n e^{cx}$ et $\frac{\mathrm{d}^n}{\mathrm{d}x^n}\left(\sin x\right) = \sin\left(x + \frac{n\pi}{2}\right)$ ; et coefficientes polynomii $f$ svnt nvmeri $\frac{f^{(k)}(0)}{k!}$ .
  Vtitvr: 8. Potentiae, polynomia et fvnctiones rationales, 13. Compositiones vsitatae, 19. Exponentialis et logarithmvs basis a, 20. Potentiae exponentis realis et variabilis, 23. Derivatae svccessivae. Nititvr: D (Hereditas Algebrae et Geometriae). Demonstratvr in: Derivatae svccessivae et differentiale.
3. Regvla Leibnitiana. Si $f$ et $g$ in intervallo aperto $n$ vicibvs derivabiles svnt, etiam $fg$ est, et $(fg)^{(n)} = \sum_{k = 0}^{n} \binom{n}{k} f^{(k)} g^{(n - k)}$ .
  Vtitvr: 6. Combinatio linearis et prodvctvm, 23. Derivatae svccessivae. Nititvr: D (Hereditas Algebrae et Geometriae). Demonstratvr in: Derivatae svccessivae et differentiale.
4. Differentiale. Si $f$ in $a$ derivabilis est, incrementvm $\Delta y = f(a + \Delta x) - f(a)$ satisfacit $\Delta y = \mathrm{d}y + \operatorname{o}(\Delta x)$ , cvm $\mathrm{d}y = f'(a) \, \Delta x$ , nec vlla alia fvnctio linearis ipsivs $\Delta x$ ei satisfacit; si $f'(a) \neq 0$ , $\Delta y \sim \mathrm{d}y$ ; et si $w = f(u)$ , tvm $\mathrm{d}w = f'(u) \, \mathrm{d}u$ .
  Vtitvr: 3. Tres characterismi, 6. Combinatio linearis et prodvctvm, 7. Inversvm et qvotiens, 12. Regvla catenae. Nititvr: B (Hereditas vnitatis 1). Demonstratvr in: Derivatae svccessivae et differentiale.
5. Ratio relativa constans. Si $y(t) = y_0 e^{kt}$ , cvm $y_0 > 0$ et $k \neq 0$ , ratio mvtationis relativa $\frac{y'(t)}{y(t)}$ aeqvalis est ipsi $k$ qvoqve instanti, et $y(t + T) = e^{kT} y(t)$ : qvantitas dvplicatvr, vel dimidiatvr, qvoqve intervallo temporis longitvdinis $\frac{\ln 2}{\abs{k}}$ .
  Vtitvr: 19. Exponentialis et logarithmvs basis a. Nititvr: D (Hereditas Algebrae et Geometriae). Demonstratvr in: Rationes mvtationis et rationes mvtationis conexae.
6. Signvm derivatae in pvncto. Si $f$ in intervallo aperto $I$ definita est, in pvncto $a$ ipsivs $I$ derivabilis est et $f'(a) > 0$ , exsistit $\delta > 0$ ita vt, pro omni $x$ ipsivs $I$ , $f(x) < f(a)$ si $a - \delta < x < a$ , et $f(x) > f(a)$ si $a < x < a + \delta$ ; non colligitvr $f$ prope $a$ crescentem esse. Hinc celeritas crescit cvm velocitas et acceleratio idem signvm habent.
  Vtitvr: 1. Definitio derivatae, 6. Combinatio linearis et prodvctvm. Nititvr: C (Conservatio signi). Demonstratvr in: Rationes mvtationis et rationes mvtationis conexae.
7. Rationes mvtationis conexae. Si $u$ in $t_0$ derivabilis est et $y(t) = g(u(t))$ , cvm $g$ in $u(t_0)$ derivabili, tvm $y'(t_0) = g'(u(t_0)) \, u'(t_0)$ ; et dvae expressiones qvae qvoqve instanti intervalli aeqvales et in $t_0$ derivabiles svnt eandem derivatam in $t_0$ habent.
  Vtitvr: 12. Regvla catenae. Demonstratvr in: Rationes mvtationis et rationes mvtationis conexae.

Ad applicationes. Catena ibi consistit vbi derivata obiectvm stvdii esse desinit et instrvmentvm fit: pars proxima a theoremate Weierstrassii de valoribvs extremis incipit; theorema Fermatii per signvm derivatae in pvncto demonstrat, et ex eo theorema Rollii et theorema valoris medii, qvae signvm derivatae in monotoniam convertvnt et regvlam Hospitalii, extrema, concavitatem et margines qvibvs differentiale caret praebent; et in methodo Newtoni et Raphsoni desinit, qvae cvrvae rectam tangentem svam svbstitvit.