Höhere Ableitungen

Duschek, Adalbert

Höhere Ableitungen

Adalbert Duschek
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Adalbert Duschek: Technischen Hochschule Wien

Chapter § 15 in Vorlesungen über höhere Mathematik, 1956, pp 166-171 from Springer

Abstract: Zusammenfassung Ist y = f(x) in einem Intervall J differenzierbar, dann ist die Ableitung % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiaacE % cacqGH9aqpdaWcaaqaaiaadsgacaWG5baabaGaamizaiaadIhaaaGa % eyypa0JaamOzaiaacEcadaqadaqaaiaadIhaaiaawIcacaGLPaaaaa % a!41A2! $$y' = \frac{{dy}}{{dx}} = f'\left( x \right)$$ wieder eine Funktion von x und man kann nach ihrer Ableitung fragen. Existiert der Grenzwert % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci % GGSbGaaiyAaiaac2gaaSqaaiaadIgacqGHsgIRcaaIWaaabeaakmaa % laaabaGaamOzaiaacEcadaqadaqaaiaadIhacqGHRaWkcaWGObaaca % GLOaGaayzkaaGaeyOeI0IaamOzaiaacEcadaqadaqaaiaadIhaaiaa % wIcacaGLPaaaaeaacaWGObaaaiabg2da9iaadAgacaGGNaGaai4jam % aabmaabaGaamiEaaGaayjkaiaawMcaaiaacYcaaaa!4F09! $$\mathop {\lim }\limits_{h \to 0} \frac{{f'\left( {x + h} \right) - f'\left( x \right)}}{h} = f''\left( x \right),$$ so nennt man ihn die zweite Ableitung oder Ableitung zweiter Ordnung von f(x) an der Stelle x und schreibt % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiaacE % cacaGGNaGaeyypa0JaamOzaiaacEcacaGGNaWaaeWaaeaacaWG4baa % caGLOaGaayzkaaGaeyypa0ZaaSaaaeaacaWGKbaabaGaamizaiaadI % haaaWaaSaaaeaacaWGKbGaamyEaaqaaiaadsgacaWG4baaaiabg2da % 9maalaaabaGaamizamaaCaaaleqabaGaaGOmaaaakiaadMhaaeaaca % WGKbGaamiEamaaCaaaleqabaGaaGOmaaaaaaGccqGH9aqpdaWcaaqa % aiaadsgadaahaaWcbeqaaiaaikdaaaGccaWGMbWaaeWaaeaacaWG4b % aacaGLOaGaayzkaaaabaGaamizaiaadIhadaahaaWcbeqaaiaaikda % aaaaaOGaaiilaaaa!568C! $$y'' = f''\left( x \right) = \frac{d}{{dx}}\frac{{dy}}{{dx}} = \frac{{{d^2}y}}{{d{x^2}}} = \frac{{{d^2}f\left( x \right)}}{{d{x^2}}},$$ wobei das Leibnizsche Symbol % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca % WGKbWaaWbaaSqabeaacaaIYaaaaOGaamyEaaqaaiaadsgacaWG4bWa % aWbaaSqabeaacaaIYaaaaaaaaaa!3BAD! $$\frac{{{d^2}y}}{{d{x^2}}} $$ aus % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca % WGKbaabaGaamizaiaadIhaaaWaaSaaaeaacaWGKbGaamyEaaqaaiaa % dsgacaWG4baaaaaa!3CB0! $$\frac{d}{{dx}}\frac{{dy}}{{dx}} $$ durch formale Multiplikation entsteht. Im Nenner eines Differentialquotienten bedeutet somit % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiaadI % hadaahaaWcbeqaaiaaikdaaaGccqGH9aqpdaqadaqaaiaadsgacaWG % 4baacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGaeyypa0Jaam % izaiaadIhacaGGUaGaamizaiaadIhacaGG7aaaaa!4478! $$d{x^2} = {\left( {dx} \right)^2} = dx.dx;$$ man läßt also zur Vereinfachung der Schreibweise die Klammern weg, hat aber dieses Symbol zu unterscheiden von dem in derselben Weise geschriebenen Differential der Funktion x2 % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiaadI % hadaahaaWcbeqaaiaaikdaaaGccqGH9aqpcaWGKbWaaeWaaeaacaWG % 4bWaaWbaaSqabeaacaaIYaaaaaGccaGLOaGaayzkaaGaeyypa0JaaG % OmaiaadIhacaWGKbGaamiEaiaac6caaaa!438C! $$d{x^2} = d\left( {{x^2}} \right) = 2xdx.$$

Date: 1956
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DOI: 10.1007/978-3-7091-3556-3_16

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