Regvla catenae ad derivandam compositionem fvnctionvm

Svmmarivm

Haec lectio regvlam catenae demonstrat: si $g$ in $a$ derivabilis est et $f$ in $g(a)$ , compositio $f \circ g$ in $a$ derivabilis est, eivsqve derivata est prodvctvm $f'(g(a)) \, g'(a)$ . Demonstratio characterismo Carathéodorii vtitvr, qvi per nihil dividit; lectio explicat cvr argvmentvm qvod per $g(x) - g(a)$ dividit non svfficiat et qvo casv tamen procedat. Regvla adhibita, compositiones potentiarvm, radicvm, fvnctionvm rationalivm et trigonometricarvm pavcis lineis derivantvr, qvae regvlis antea traditis longas evolvtiones postvlabant avt extra facvltatem erant. Lectio etiam dominivm compositionis cvrat, qvod formvla derivatae non ostendit, et regvlam in notatione Leibnitiana exhibet, vbi simplificatio fractionvm esse videtvr, qvamqvam non est.

Proposita discendi

Regvlam catenae characterismo Carathéodorii demonstrare, et explicare cvr demonstratio qvae per $g(x) - g(a)$ dividit deficiat cvm $g'(a) = 0$ .
Compositiones dvarvm plvrivmve fvnctionvm notarvm derivare, vnamqvamqve derivatam in pvncto sibi respondenti aestimando.
Dominivm compositionis ante derivationem determinare, et agnoscere hypotheses regvlae svfficientes esse, non avtem necessarias.
Regvlam catenae in notatione Leibnitiana scribere et legere.

Derivata compositionis

Regvlis lectionis de regvlis derivandi et formvlis lectionis de derivatis fvnctionvm elementarivm derivari potest $\left(2x^2 + 1\right)^{12}$ , sed tantvm potentiam in polynomivm gradvs $24$ evolvendo, avt regvlam prodvcti itervm atqve itervm adhibendo. Neqve adhvc derivari possvnt $\sin(\cos x)$ avt $\sqrt{1 + x^2}$ : nvlla regvla dicit qvid faciendvm sit de fvnctione alteri applicata. Ambae expressiones compositiones svnt: $\left(2x^2 + 1\right)^{12}$ est potentia $t^{12}$ in $t = 2x^2 + 1$ aestimata, et $\sin(\cos x)$ est sinvs in $t = \cos x$ aestimatvs.

Dvo casvs particvlares iam noti svnt. Problema solvtvm 2 lectionis de fvnctionibvs elementaribvs ostendit $f(cx + b)$ in $a$ derivatam $c f'(ca + b)$ habere: fvnctio interior linearis est, declivitate $c$ , et derivata exterioris in pvncto $ca + b$ aestimatvr, non in $a$ . Et problema propositvm 4 lectionis de regvlis derivandi per indvctionem obtinvit $\left(f(x)\right)^n$ in $a$ derivatam $n \left(f(a)\right)^{n - 1} f'(a)$ habere. In vtroqve casv derivata prodvctvm est: derivata fvnctionis exterioris, in valore interioris aestimata, dvcta in derivatam interioris. Idea est rationvm mvtationis: si $t = g(x)$ mvtatvr $g'(a)$ vicibvs celerivs qvam $x$ , et $f(t)$ mvtatvr $f'(g(a))$ vicibvs celerivs qvam $t$ , tvm $f(g(x))$ mvtatvr $f'(g(a)) \, g'(a)$ vicibvs celerivs qvam $x$ .

TheoremaRegvla catenae

Sint $g$ fvnctio in $A$ definita, derivabilis in pvncto $a$ copiae $A$ qvod pvnctvm accvmvlationis copiae $A$ est, et $f$ fvnctio in copia $B$ definita qvae valores fvnctionis $g$ continet, derivabilis in $b = g(a)$ . Tvm $f \circ g$ in $a$ derivabilis est, et

(f \circ g)'(a) = f'(g(a)) \, g'(a)

Dvo incrementa characterismo Carathéodorii factorizantvr et deinde conivngvntvr: incrementvm fvnctionis $f$ in $t = g(x)$ aestimatvr, et incrementvm fvnctionis $g$ in locvm $t - b$ svbstitvitvr. Figvra ostendit, cvm $g(x) = x^2$ , $f(t) = \sin t$ et $a = 1$ , rectam secantem et rectam tangentem fvnctionis $g$ , easdem fvnctionis $f$ in $b = 1$ et, ad extremvm, easdem fvnctionis $\sin\left(x^2\right)$ , cvivs declivitas in $1$ est prodvctvm $\cos 1 \cdot 2$ .

Demonstratio

$g(x) - g(a) = \dato{psi}{\psi(x)(x - a)}, \quad x \in A, \qquad f(t) - f(b) = \dato{fi}{\varphi(t)(t - b)}, \quad t \in B$
Per theorema de characterismis derivatae, ex lectione de derivata vt limite, exstant $\psi \colon A \to \mathbb{R}$ , continva in $a$ cvm $\psi(a) = g'(a)$ , et $\varphi \colon B \to \mathbb{R}$ , continva in $b$ cvm $\varphi(b) = f'(b)$ , ita vt valeat:
$f(g(x)) - f(g(a)) = \varphi(g(x)) \left(g(x) - g(a)\right) = \resaltar{\varphi(g(x)) \, \psi(x)}(x - a)$
Sit $x$ elementvm copiae $A$ . Nvmervs $t = g(x)$ ad $B$ pertinet, et altera aeqvalitas pro eo valet, sive aeqvalis est $b$ sive non; deinde $t - b = g(x) - g(a)$ per primam factorizationem svbstitvitvr.
$\gamma(a) = \varphi(g(a)) \, \psi(a) = \varphi(b) \, \psi(a) = \resaltar{f'(g(a)) \, g'(a)}$
Cvm $g$ in $a$ derivabilis sit, in $a$ continva est, per theorema «derivabilis ergo continva»; et $\varphi$ in $b = g(a)$ continva est. Per theorema de compositione fvnctionvm continvarvm, ex lectione de limite et continvitate, $\varphi \circ g$ in $a$ continva est, et factor $\gamma = (\varphi \circ g) \, \psi$ item, per regvlam prodvcti operationvm cvm fvnctionibvs continvis. Cvm $a$ pvnctvm accvmvlationis copiae $A$ sit, qvae dominivm fvnctionis $f \circ g$ est, characterismvs Carathéodorii derivatam praebet.

Si $g$ in tota $A$ derivabilis est et $f$ in tota $B$ , regvla in omni pvncto accvmvlationis copiae $A$ valet, et vt aeqvalitas inter fvnctiones scribitvr: $(f \circ g)' = (f' \circ g) \, g'$ . Derivata fvnctionis exterioris in $g(x)$ aestimatvr, non in $x$ ; hic error in regvla adhibenda freqventissimvs est. Commodvm demonstrationis in secvndo eivs articvlo positvm est: factorizatio fvnctionis $f$ etiam in $t = b$ valet, vbi ambo eivs membra nvlla svnt, ita vt nihil referat vtrvm $g$ valorem svvm $g(a)$ prope $a$ iteret necne.

In reliqva parte hvivs lectionis

Dominivm compositionis
Cvr dividere non svfficiat
Compositiones fvnctionvm notarvm
Regvla catenae in notatione Leibnitiana
Problemata solvta
Rectae tangentes horizontales compositionis triplicis
Derivare ex tabvla valorvm
Hypotheses svfficientes, non necessariae
Fvnctio circvmscripta cvm derivata non circvmscripta
Fvnctio secvm composita
Problemata proposita
Derivata fvnctionis paris
Identitatem derivare
Valor absolvtvs extra nihila
Ivnctvra per regvlam catenae
Fvnctio exterior qvae derivabilis non est