Variabilis in exponente et derivatio logarithmica

Svmmarivm

Haec lectio fvnctiones derivat qvae variabilem in exponente habent. Omnes ad exponentialem natvralem redigvntvr: $a^x = e^{x \ln a}$ , $x^r = e^{r \ln x}$ et $\left(f(x)\right)^{g(x)} = e^{g(x) \ln f(x)}$ , ita vt derivata fvnctionis $e^x$ , derivata logarithmi et regvla catenae svfficiant. Sic obtinentvr derivatae exponentialis et logarithmi cvivsvis basis, potentiarvm exponentis realis et potentiarvm qvarvm et basis et exponens variantvr, vt $x^x$ vel $x^{\sin x}$ . Derivata fvnctionis $\ln \abs{f(x)}$ , qvae est $\frac{f'(x)}{f(x)}$ , prodvcta, qvotientes et potentias in svmmas convertit: haec est derivatio logarithmica, qvam lectio confirmat neqve svpponit qvod demonstrare vvlt. In fine, recta tangens logarithmi per originem transiens decernit vter maior sit, $\pi^e$ an $e^\pi$ , et explicat cvr exitvs a nvmero $e$ pendeat.

Proposita discendi

Exponentialem et logarithmvm cvivsvis basis, et compositiones eorvm, derivare, ab aeqvalitate $a^x = e^{x \ln a}$ incipiendo.
Potentias exponentis realis et potentias basis et exponentis variabilivm derivare, eas vt exponentiales basis $e$ scribendo, in dominio vbi basis positiva est.
Derivationem logarithmicam ad prodvcta, qvotientes et potentias adhibere, antea confirmando fvnctionem derivabilem esse nec in pvncto evanescere.
Potentias vt $\pi^e$ et $e^\pi$ comparare per rectam tangentem logarithmi et inaeqvalitatem fvndamentalem exponentialis.

Exponentialis et logarithmvs cvivsvis basis

Lectio de derivatis fvnctionvm elementarivm demonstravit $e^x$ svam ipsivs derivatam esse et derivatam fvnctionis $\ln x$ esse $\frac{1}{x}$ , et nvntiavit derivatam fvnctionis $a^x$ esse $a^x \ln a$ . Non opvs est calcvlvm per qvotientem incrementalem iterare: omnis basis positiva potentia est nvmeri $e$ . Per definitionem logarithmi natvralis, ex lectione de fvnctione logarithmo, $e^{\ln y} = y$ pro omni $y > 0$ ; et per legem logarithmi potentiae, ex eadem lectione, $\ln\left(a^x\right) = x \ln a$ . Per regvlam catenae reliqvvm statim patet.

TheoremaDerivatae exponentialis et logarithmi basis a

Sit $a > 0$ , cvm $a \neq 1$ . Exponentialis $a^x$ derivabilis est in $\mathbb{R}$ , logarithmvs $\log_a x$ derivabilis est in $\ioo{0}{\infty}$ , et fvnctio $\log_a \abs{x}$ derivabilis est in $\mathbb{R} \setminus \{0\}$ :

\frac{\mathrm{d}}{\mathrm{d}x}\left(a^x\right) = a^x \ln a, \qquad \frac{\mathrm{d}}{\mathrm{d}x}\left(\log_a x\right) = \frac{1}{x \ln a} \quad (x > 0), \qquad \frac{\mathrm{d}}{\mathrm{d}x}\left(\log_a \abs{x}\right) = \frac{1}{x \ln a} \quad (x \neq 0)

Exponentialis basis $a$ per exponentialem natvralem scribitvr et regvla catenae adhibetvr; logarithmvs basis $a$ mvltiplvm constans logarithmi natvralis est. Figvra ostendit $2^x$ , $e^x$ et $4^x$ cvm rectis tangentibvs in pvncto $(0, 1)$ , declivitatvm $\ln 2$ , $1$ et $\ln 4$ , et in fine rectam tangentem fvnctionis $\log_2 x$ in $(1, 0)$ , reflexionem tangentis fvnctionis $2^x$ .

Demonstratio

$a^x = e^{\ln\left(a^x\right)} = \dato{base}{e^{x \ln a}}$
Sit $x$ nvmervs realis. Nvmervs $a^x$ positivvs est; per definitionem logarithmi natvralis et legem logarithmi potentiae, ex lectione de fvnctione logarithmo:
$\frac{\mathrm{d}}{\mathrm{d}x}\left(e^{x \ln a}\right) = e^{x \ln a} \cdot \ln a = \resaltar{a^x \ln a}$
Fvnctio interior $x \ln a$ linearis est, derivatae $\ln a$ , et exterior $e^t$ derivabilis est in toto $\mathbb{R}$ , per theorema de derivata exponentialis natvralis. Per regvlam catenae, ex lectione de regvla catenae:
$\frac{\mathrm{d}}{\mathrm{d}x}\left(\log_a \abs{x}\right) = \frac{1}{\ln a} \cdot \frac{1}{x} = \resaltar{\frac{1}{x \ln a}}$
Per theorema mvtationis basis, ex lectione de fvnctione logarithmo, $\log_a y = \frac{\ln y}{\ln a}$ pro omni $y > 0$ , cvm $\ln a \neq 0$ qvia $a \neq 1$ . Per theorema de derivata logarithmi natvralis, qvod pro $\ln x$ et pro $\ln \abs{x}$ valet, et per theorema de derivata combinationis linearis, ex lectione de regvlis derivandi:

Derivata fvnctionis $a^x$ fvnctioni proportionalis est, cvm constante $\ln a$ , qvae positiva est si $a > 1$ et negativa si $0 < a < 1$ ; propter iniectivitatem logarithmi, aeqvalis est $1$ tantvm si $a = e$ . Cvm $a = 1$ , fvnctio $1^x$ constans est et formvla valere pergit, qvia $\ln 1 = 0$ . In $x = 0$ , definitio derivatae limitem $\frac{a^h - 1}{h} \to \ln a$ praebet, qvi est corollarivm de limitibvs notabilibvs logarithmi et potentiarvm, ex lectione de continvitate fvnctionvm elementarivm: derivata fvnctionis $a^x$ in $0$ iam nota erat, et lex $a^{x + h} = a^x a^h$ eam ad omnia pvncta transfert.

CorollarivmCompositiones cvm exponentiali et logarithmo

Sit $u$ fvnctio in $A$ definita, derivabilis in pvncto $a$ copiae $A$ qvod pvnctvm accvmvlationis est copiae $A$ , et sit $b > 0$ , cvm $b \neq 1$ . Tvm $e^{u(x)}$ et $b^{u(x)}$ derivabiles svnt in $a$ . Si $u(a) \neq 0$ , fvnctiones $\ln \abs{u(x)}$ et $\log_b \abs{u(x)}$ derivabiles svnt in $a$ , in svo dominio; et si $u(a) > 0$ , etiam $\ln u(x)$ derivabilis est, eadem derivata ac $\ln \abs{u(x)}$ . Derivatae in $x = a$ svnt:

\frac{\mathrm{d}}{\mathrm{d}x}\left(e^{u(x)}\right) = e^{u(x)} u'(x), \qquad \frac{\mathrm{d}}{\mathrm{d}x}\left(b^{u(x)}\right) = b^{u(x)} \ln b \cdot u'(x)

\frac{\mathrm{d}}{\mathrm{d}x}\left(\ln \abs{u(x)}\right) = \frac{u'(x)}{u(x)}, \qquad \frac{\mathrm{d}}{\mathrm{d}x}\left(\log_b \abs{u(x)}\right) = \frac{u'(x)}{u(x) \ln b}

Dvae primae svnt regvla catenae; dvae vltimae, eivs forma in dominio compositionis, qvia logarithmvs in $0$ definitvs non est.

Demonstratio

$\frac{\mathrm{d}}{\mathrm{d}x}\left(e^{u(x)}\right) = e^{u(x)} u'(x), \qquad \frac{\mathrm{d}}{\mathrm{d}x}\left(b^{u(x)}\right) = \resaltar{b^{u(x)} \ln b \cdot u'(x)}$
Fvnctiones exteriores $e^t$ et $b^t$ derivabiles svnt in toto $\mathbb{R}$ , qvod valores fvnctionis $u$ continet; regvla catenae dvas primas derivatas praebet, in $x = a$ .
$\frac{\mathrm{d}}{\mathrm{d}x}\left(\ln \abs{u(x)}\right) = \frac{1}{u(x)} \cdot u'(x) = \resaltar{\frac{u'(x)}{u(x)}}$
Fvnctio $\ln \abs{t}$ in $B = \mathbb{R} \setminus \{0\}$ definita est, qvae copia intervallvm centri $u(a)$ et radii $r = \abs{u(a)}$ continet, et derivabilis est in $u(a)$ , derivata $\frac{1}{u(a)}$ . Per corollarivm de regvla catenae in dominio compositionis, $a$ pvnctvm accvmvlationis est dominii fvnctionis $\ln \abs{u(x)}$ , et derivata in $x = a$ est:
$\frac{\mathrm{d}}{\mathrm{d}x}\left(\log_b \abs{u(x)}\right) = \frac{1}{\ln b} \cdot \frac{u'(x)}{u(x)} = \resaltar{\frac{u'(x)}{u(x) \ln b}}$
Si $u(a) > 0$ , idem corollarivm, cvm $B = \ioo{0}{\infty}$ et $r = u(a)$ , derivatam fvnctionis $\ln u(x)$ praebet, qvae eadem est. Pro basi $b$ per constantem $\ln b$ dividitvr, per mvtationem basis.

Sic derivata fvnctionis $2^{\sin x}$ est $2^{\sin x} \ln 2 \cdot \cos x$ , et derivata fvnctionis $\ln\left(x^2 + 1\right)$ est $\frac{2x}{x^2 + 1}$ in toto $\mathbb{R}$ . Lectio de regvla catenae fvnctionem $\ln(\cos x)$ in svo dominio derivavit, vbi $\cos x > 0$ , et $-\tan x$ obtinvit; valor absolvtvs effectvm extendit: $\ln \abs{\cos x}$ definita est vbi $\cos x \neq 0$ , et eivs derivata est $-\tan x$ in tota hac copia, qvae dominivm tangentis est.

In reliqva parte hvivs lectionis

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Derivatio logarithmica
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Problemata proposita
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