Differentialrechnung: Aufgaben Jörg Gayler, Lubov Vassilevskaya
ii Inhaltsverzeichnis. Erste Ableitung der elementaren Funktionen......................... Ableitungsregeln...................................... 3. Einige Regeln........................................ 3.. Rechenregeln mit Potenzen............................. 3.. Rechenregeln mit Wurzeln............................. 3.3. Rechenregeln mit Logarithmen........................... 3.4. Trigonometrische Formeln............................. 3 4. Aufgaben zur Bestimmung von Ableitungen........................ 4 4.. Summenregel................................... 4 4.. Produktregel.................................... 5 4.3. Quotientenregel.................................. 6 4.4. Kettenregel..................................... 7 4.4.. Ableitungen von Logarithmusfunktionen................ 9 5. Logarithmische Differentiation............................... 6. Ableitungsregeln: Lösungen................................ 3 6.. Summenregel.................................... 3 6.. Produktregel.................................... 5 6.3. Quotientenregel................................... 6 6.4. Kettenregel..................................... 8 6.4.. Ableitungen von Logarithmus- und Eponentialfunktionen...... 4 7. Logarithmische Differentiation............................... 8
Ableitungsregeln. Erste Ableitung der elementaren Funktionen C = 0 C = const, n = n n n R log a = ln a a > 0, ln = > 0 e = e, a = ln a a sin = cos, cos = sin tan = cos, cot = sin arcsin =, arccos = arctan = +, arccot = + sinh = cosh, cosh = sinh tanh = cosh, coth = sinh arsinh = +, arcosh = artanh =, arcoth =. Ableitungsregeln Faktorregel: y = C f C = const, y = C f Summenregel: y = f + f +... + f n, y = f + f +... + f n Produktregel: y = u v, y = u v + u v Quotientenregel: y = u v, y = u v u v v Kettenregel: y = F u, y = dy d = dy du du d
Einige Regeln 3. Einige Regeln 3.. Rechenregeln mit Potenzen a n a m = a n + m, a n b n = a b n, a n a m = an m, a n b n = a b n a 0 =, a n = a n, an = a n a n m = a n m 3.. Rechenregeln mit Wurzeln Wurzeln als Potenzen mit gebrochenen Eponenten a m n = n a m, n a n b = n a b, n n n a b = a b m n n a = m a = m n a 3.3. Rechenregeln mit Logarithmen log a y = log a + log a y log a = log y a log a y log a r = r log a log a = log a
Einige Regeln 3.4. Trigonometrische Formeln sin α + cos α = sin α = sin α, cos α = cos α π sin α = cos α, π cos α = sin α sin α = sin π α, cos α = cos π α sin α = sin α cos α, cos α = cos α sin α sin α ± β = sin α cos β ± cos α sin β, cos α ± β = cos α cos β sin α sin β α + β α β α + β α β sin α + sin β = sin cos, sin α sin β = cos sin α + β α β α + β α β cos α + cos β = cos cos, cos α cos β = sin sin sin α sin β = sin α cos β = [ cos α β cos α + β ], cos α cos β = [ sin α β + sin α + β ] [ cos α β + cos α + β ] sin α = cos α, cos α = + cos α sin 3 α = 4 3 sin α sin 3α, cos3 α = cos3α + 3 cos α 4
Aufgaben zur Bestimmung von Ableitungen 4. Aufgaben zur Bestimmung von Ableitungen 4.. Summenregel Bestimmen Sie die erste Ableitung folgender Funktionen a, b, c, m, n R: A Beispiel: f = 3 3 = 3 3, f = 3 3 3 = 6 + 6 3 = 6 + 3 a f = 3 4 3 + 6, g = 5 3 + 5 b f = 3 + 4 3, g = 3 3 + 3 4 c f = m 5 n, g = a m + n b cm n A Beispiel : Beispiel : f = 5 + 3 = + 5 + + 3 = 5 + 3 4 f = 5 5 4 3 4 3 = 5 6 5 4 3 7 3 = 5 5 f = 4 3 = /4 /3 = 4 3 = f = = = + 4 3 3 a f = +, g = + 3 + 3, h = 3 + 5 b f = + +, g = 3 + 5 + 3 6 c f = 3, g = 3 4, h = 3 3 6
Aufgaben zur Bestimmung von Ableitungen A3 Beispiel: f = 3 3 3 = 3 3 /3 = 3 3 4/3 = 3 4/9 = 3 9 = 7 f = 7 7 = 7 5 7 = 7 5 7 = 7 7 5 a f = b f =, g =, g = 3, h = 3 4 3 5, h = 3 4 A4 a f = sin + tan + ln, g = 3 sin 7 cos b f = tan cot, g = sinh + 3 cosh A5 Bestimmen Sie die drei ersten Ableitungen folgender Funktionen: a f = 4 + + e b f = sin + cos c f = ln + e d f = sinh + cosh e f = n + 4 + 3 A6 4.. Produktregel Folgende Funktionen sind unter Verwendung der Produktregel zu differenzieren: a f = +, g = + b f = + + 3, g = + + c f = +, g = +
Aufgaben zur Bestimmung von Ableitungen A7 a f = ln, g = + ln b f = cos, g = cot, h = tan c f = 3 + + e sin 4.3. Quotientenregel Folgende Funktionen sind unter Verwendung der Quotientenregel zu differenzieren: A8 a f =, g = +, h = 3 5 b f = +, g = +, h = + c f = + +, g =, h = + 3 4 d f =, g = +, h = 3 4 e f =, g = +, h = 3 + 4 A9 a f =, g = 3, h = 3 4 b f = c f = + + +, g = 3 + +, h = + 3 + 5 +, g = + 4 + 4 4 d f = 6 + 9 8, g = + + 3 e f = + + + 3, g = 8 + 8 3 f f = + +, g =, h =, h = 4 + + 9 4 9 + g f = + + +, g = +, h = +
Aufgaben zur Bestimmung von Ableitungen 4.4. Kettenregel Folgende Funktionen sind unter Verwendung der Kettenregel zu differenzieren: A0 a f = 6, g = + 5, h = 4 3 4 b f = 4 5, g = 3 3 + 6 9, h = 4 3 3 c f = 4 4, g = 3 3 7, h = 4 3 3 A a f = 6, g = 3 + 3 3, h = 3 + + ln 4 b f = + 6, g = 3 + 4, h = 3 3 3 c f = 3 3, g = 3 4 + 5 +, h = 5 3 A f = 4, g = + 3 5 +, h = 3 3 + 3 + 4 7 A3 f = +, g = 5 +, h = + 4 A4 f = +, g = 3 7, h = + 4 A5 a f = sin 3, g = sin 4 +, h = sin 4 b f = cos, g = cos3 +, h = cos 3 4 c f = sin, g = sin 3, h = sin 3 + 4 d f = cos, g = cos 5, h = cos 5 3 +
Aufgaben zur Bestimmung von Ableitungen A6 a f = sin, g = sin 5, h = 5 sin, p = 5 sin 3 b f = cos, g = cos, h = 4 cos, p = 4 cos + 6 c f = d f = sin 5, g =, h = sin 5 cos, g = cos 3, h = sin sin 5 +, p = sin cos sin, p = sin 3 + cos A7 a f = sin 3, g = sin 3, h = sin 3 b f = cos 4, g = cos 4, h = cos 4 c f = sin +, g = sin +, h = d f = cos 3, g = cos 3 7, h = sin 3 + cos 4 + A8 a f = sin sin + π, g = sin sin + π b f = cos cos + π, g = cos cos + π A9 a f = sin cos, g = sin 3 cos + π b f = sin cos, g = sin +, h = sin + c f = sin cos 3, g = sin cos A0 f = e, g = e 3, h = e, p = e + 4 A a f = 3 e, g = e, h = e 3, e p = b f = e, g = + 3 e, h = e, p = 3 e
Aufgaben zur Bestimmung von Ableitungen A a f = cos e, g = cos 3 e, h = sin e b f = sin + cos e, g = e sin, h = e sin 4.4.. Ableitungen von Logarithmusfunktionen A3 Regeln: ln u = u u, eu = e u u, a u = a u ln a u Beispiel : f = ln 4 + 3, f = ln 4 + 3 + ln 4 + 3 u = 4 + 3, ln 4 + 3 = ln u = u u f 4 = ln 4 + 3 + 4 + 3 = 4 + 3 4 + 3 = 4 4 + 3 Beispiel : f = ln + 3 = 3 ln +, f = 3 + a f = ln, g = ln, h = + ln b f = ln 5, g = ln 3, h = ln 3 + c f = + ln +, g = ln, h = ln d f = ln 4 + 4, g = ln, h = ln 3 8 e f = ln + 4, g = ln 5, h = ln + 3 3 A4 Beispiel : Beispiel : f = ln 5 = ln 5 = 5 ln, f = 5 f = ln + = ln + = ln +, f = + a f = ln, g = ln 3, h = ln n b f = ln +, g = ln +, h = ln + + c f = ln + +, g = ln
Aufgaben zur Bestimmung von Ableitungen A5 Beispiel : f = ln 3 + 3 = ln + 3 f = 3 = + 3 Beispiel : f = ln 3 = 3 ln 5 3 + 3 3 = 9 3 ln 9 = 3 = + 3 3 ln [3 3 + ] = ln ln + 3 = 3 f = 3 [ln 3 + ln 3 + ] [ 3 + ] = 3 + 3 9 a f = ln, g = ln + + +, h = ln + 4 b f = ln, g = ln + 3 + + +, h = ln 5 4 + + + c f = ln, g = ln, h = ln A6 a f = ln cos, g = ln sin, h = ln tan b f = ln cos 3, g = ln sin 5, h = ln tan c f = ln cos, g = ln sin 3, h = ln tan + 4 d f = ln cos 3, g = ln sin 3, h = ln tan + 4 A7 a f = ln cos, g = ln + sin, h = ln tan b f = ln 3 cos, g = ln 4 + sin, h = ln 3 tan A8 a f = ln, g = 3 ln, h = + ln A9 f = + ln ln, g = ln, h = lnsin
Logarithmische Differentiation 5. Logarithmische Differentiation Bei manchen Differentiationsaufgaben ist es sinnvoll, die Funktionsgleichung y = f vor dem Differenzieren zu logarithmieren. Das bietet sich immer dann an, wenn sich die Funktionsgleichung durch Anwendung der Logarithmengesetze wesentlich vereinfachen lässt. A30 Benutzen Sie die Methode der logarithmischen Differentiation, um die Produktregel für die Funktion f = u v und die Quotientenregel für die Funktion f = u /v zu beweisen. A3 A3 A33 A34 Bestimmen Sie die Ableitungen folgender Funktionen durch Logarithmische Differentiation Beispiel : a y = 5, y = 3 4, y = 7 3 b y = e, y = e e c y = e, y = e a y = e, y = 4 3 e b y = +, y = + 7 c y = e 3 4 + a y = + 4, b y =, c y = y = +, ln y = ln ln + = + = y y = + = y = y = + = + + + = + =
Logarithmische Differentiation Beispiel : + y = 3 + 3 = + + ln y = + 3 ln = ln + 3 + ln ln + = + = 3 ln + + ln ln + ln y = y y = 3 + + + y = y 3 + + = + = 3 + + + 3 + + Aufgabe: a y = +, b y = +, c y = + 3 + 4, d y = + A35 a y = sin 3, b y = 3 cos, c y = tan e
Ableitungsregeln: Lösungen 6. Ableitungsregeln: Lösungen 6.. Summenregel L a f = 3 4 3 + 6, f = 3 6 g = 5 3 + 5, g = 5 0 3 b f = 3 + 4 3, f = 3 8 3 + 6 4 g = 3 3 + 3 4 = + 3, g = 3 + 3 4 c f = m 5 n, f = m m 5 n n g = a m + n b cm n, g = a m m + n b n + c n m m n L a f = +, f = + g = + 3 3 + = 3/ + 4/3 + 8/3, g = 3 / + 4 3 /3 + 8 3 5/3 = 3 + 4 3 3 + 8 3 3 h = 3 + 5 = /3 + /5, h = 3 /3 + 5 4/5 = 3 3 + 5 5 4 b f = / + / + 3/, f = 3 g = 3 + 5 3 + 6 = /3 + 3/5 + 5/6, g = 3 4/3 3 5 8/5 5 6 /6 = 3 3 3 5 5 3 5 6 6 5 c f = g = h = 3 = /6, f = 6 7/6 = 3 4 = /, g = / 3 3 =, h 6 = 6 6
Ableitungsregeln: Lösungen L3 a f = g = = 3/4, f = 3 4 /4 = 3 4 4 3 = /3, g = 3 /3 = 3 3 h = 3 4 = /3, h = 3 /3 = b f = g = h = 3 3 = 7/8, f = 7 8 /8 = 7 8 8 3 5 = 5/8, g = 5 8 7/8 = 5 8 3 4 = 7/8, h = 7 8 /8 = 7 8 8 8 7 L4 a f = sin + tan + ln, f = cos + + tan + g = 3 sin 7 cos, g = 3 cos + 7 sin b f = tan cot, f = sin cos g = sinh + 3 cosh, g = cosh + 3 sinh L5 a f = 4 + 3/ + e, f = 4 3 + 3 + e f = + 3 4 + e, f = 4 b f = sin + cos, f = cos sin f = sin cos, f = cos + sin 3 + e 8 3/ c f = ln + e, f = + e f = + e, f = 3 + e d f = sinh + cosh, f = f = f = sinh + cosh e f = n + 4 + 3, f = n n n n 3 + 4 + 6
Ableitungsregeln: Lösungen 6.. Produktregel L6 a f = +, f = 4 + 3 g = + =, g = b f = + + 3, f = + + 3 8 3 g = + + = + = 4, g = 4 3 c f = + =, f = g = +, g = + L7 a f = ln, f = ln + g = + ln, g = ln + + b f = cos, f = cos sin g = cot, g = cot + cot = cot cot h = tan, h = tan + + tan, c f = 3 + + e sin, f = 3 + + e sin + 3 + + e cos
Ableitungsregeln: Lösungen L8 6.3. Quotientenregel a f =, g = +, h = 3 5, f = g = h = + 5 5 b f = +, f = g = +, h = +, g = h = c f = +, f = g = +, h = + 3 4, d f = g = h = g = h = + 4 4 + 4 3 4, f = + +, g = + 3 4, h = 3 4 + 4 e f = = + = + g = + = + = +,, f = g = 4 h = 3 + 4 = 3 + + = 3 + 7 h = 7,
Ableitungsregeln: Lösungen L9 a f =, f = 3 = 3 g = 3, g = 3 4 = 3 4 h = 3 4, h = 3 4 = 3 4 = g Die Ableitungen f, g und h kann man auch mit Hilfe der Summenregel bestimmen, indem man die Funktionen f, g und h entsprechend darstellt: f =, g = 3, h = 3 = g b f = g = +, f = + + 3 + +, g = 3 + + 5 + h = + 3 + 5 +, h = 5 + 8 + 3 + c f = + + g = + 4 + 4 4 = = h = 4 + + 9 4 9 + + = +, + + = +, = + 3 3 + 3 = + 3 3, f = g = 4 h = 3 d f = 6 + 9 8 = 3 9 = 3 3 + 3 = 3 + 3, f = 3 + 3 g = + + 3 g = + = + + = + + = + e f = + + + + 3 = + 3 = +, g = 8 + 8 3 = 3 =, f = g = +
Ableitungsregeln: Lösungen f f = +, f = + + g =, g = h =, h = + + + 3/ + = 3/ g f = + + +, f = + + + g = +, g = + h =, h 3 = + + 3/ 6.4. Kettenregel L0 a f = 6, f = 6 5 g = + 5, g = 4 + 5 h = 4 3 4, h = 6 4 3 5 b f = 4 5, f = 0 4 4 g = 3 3 + 6 9, g = 54 3 3 + 6 8 + h = 4 3 3, h = 4 4 3 3 = 4 3 7 4 c f = g = h = h = 3 4 4, f = 4 4 3 + 4 = 8 5 3 4 3 3 + 3 3 7, g = 7 3 3 6 3 + 6 3 = 4 3 3, 5 5 3 6 5 + 4 3 + + 4 = 6 0 5 4 5 + + 6
Ableitungsregeln: Lösungen L a f = 6, f = 6 5 g = 3 + 3 3, g = 3 /3 + 3 3 /3 + 3 h = 3 + + ln 4, h = 4 + /3 + ln 3 = 3 3 + 3 3 + 3 3 + 4 3 + /3 b f = + 6, f = + 6 g = 3 + 4, g = 3 4 + 4 3 + 4 h = 3 3 3, h = 3 3 /3 c f = 3 3, f = 3 3 g = 3 4 + 5 +, g = 3 h = 5 3, h = 3 5 3 4/5 4 3 + 5 4 + 5 + /3 L f = g = h = 4, f 3 = 6 + 3 + 3 5 5 +, g = 5 + 4 6 3 3 6 3 7 + 3, h = 4 3 + 3 6 3 + 6 + 7 + 4 + 4 8 L3 f = +, f = + g = 5 +, g = 5 + 4/5 + h = + 4 =, h = 3/
Ableitungsregeln: Lösungen L4 f = +, f = + + + = 3 + 4 + g = h = + 3 7, g = 4, h = 3 7 + 3 4 + 3 = 7 7 3 6 7 /3 + 4 L5 a f = sin 3, f = 3 cos 3 g = sin 4 +, g = 4 cos 4 + h = sin 4, h = cos 4 b f = cos = cos, f = sin g = cos3 +, g = 3 sin 3 + h = cos 3 4, h = 4 3 sin 3 4 c f = sin, f = cos g = sin 3, g = 3 cos 3 3 h = sin 3 + 4, h = 3 cos 3 + 4 3 + 4 d f = cos, f = sin g = cos 5, g = 5 sin 5 5 h = cos 5 3 +, h = 3 5 sin 5 3 + 5 3 + 4
Ableitungsregeln: Lösungen L6 a f = sin, f = cos sin g = sin 5, g = 5 cos 5 sin 5 h = 5 sin, h = 5 p = 5 sin 3, p = 5 cos sin 4/5 = 5 b f = cos, f = sin cos g = cos, g = h = 4 cos, h = 4 p = 4 cos + 6, p = c f = g = h = p = d f = g = h = p = sin 5, cos 5 sin 4 cos 3 sin 3 4/5 = 5 sin cos sin cos 3/4 = 4 f = 5 cos5 sin 5 sin 5, g = 5 sin sin 5 +, h = sin 4 cos 3 sin + 6 cos + 6 3/4 = cos 5 sin 5 3/ = 5 cos sin 5 + sin, p = cos sin cos, f = cos 3, g = 3 cos sin 3 +, h = sin cos sin 3 cos 3 3/ = 3 sin sin 3 + cos 3 5 sin 4 3 cos 5 sin 3 5 5 sin cos 5 + sin 5 + sin 3 cos 3 3 sin + 6 4 cos 3 + 6 3 cos cos 3 + sin 3 + sin cos, p = cos cos + sin sin cos
Ableitungsregeln: Lösungen L7 a f = sin 3, f = 3 cos 3 g = sin 3, g = 3 cos sin h = sin 3, h = 6 cos sin b f = cos 4, f = 4 3 sin 4 g = cos 4, g = 4 sin cos 3 h = cos 4, h = 8 sin cos 3 c f = sin +, f = sin + cos + + g = sin +, g = cos sin + h = sin 3 +, h = 3 sin cos sin 3 + d f = cos 3, f = 3 sin 3 cos 3 3 g = cos 3 7, g = 3 sin 3 cos 3 7 h = cos 4 +, h = cos3 sin cos 4 + L8 a f = sin sin + π = sin, f = 4 sin cos = sin4 g = sin sin + π = sin sin, g = cos sin sin cos b f = cos cos + π = cos sin = sin, f = cos = sin cos g = cos cos + π = cos sin, g = sin sin cos cos
Ableitungsregeln: Lösungen L9 a f = sin cos, f = sin cos sin 3 = sin cos sin g = sin 3 cos + π = sin 3 cos, g = 6 sin3 cos cos3 + sin 3 sin = sin3 sin sin3 6 cos cos3 b f = sin cos, f = cos g = sin +, g = + cos + h = sin +, h = cos + c f = sin cos 3, f = cos cos 3 3 sin sin 3 g = sin cos, g = cos sin = cos L0 f = e, g = e 3, h = e, f = e g = e 3 h = e = e p = e + 4, p = + 3 e + 4 L a f = 3 e, f = e 3 + g = e, g = e + h = e 3, p = e, h = e 3 3 e 4 p = e b f = e, f = e g = + 3 e, = e 3 4 e = e g = + + 3 e h = e, h = e + p = 3 e, p = 3 e
Ableitungsregeln: Lösungen L a f = cos e, f = e cos sin g = cos 3 e, g = e cos 3 3 sin 3 h = sin e, h = e cos + sin b f = sin + cos e, f = e 3 cos sin g = e sin, g = cos e sin h = e sin, h = cos e sin 6.4.. Ableitungen von Logarithmus- und Eponentialfunktionen L3 a f = ln, f = + ln g = ln, g = + ln = + ln h = + ln, h = + + + ln b f = ln 5, f = g = ln 3, g = ln 3 + h = ln 3 +, h = ln 3 + + 3 3 + c f = + ln +, f = + ln + = [ + ln + ] g = ln, g = ln + + h = ln, h = ln + = [ln + ] d f = ln 4 + 4, f = 4 4 + 4 = g = ln, g = 4 ln + 4 h = ln 3 8, h = ln 3 8 + 3 3 8 = ln 3 8 + 3 + + 4 e f = ln + 4, f = + 4 g = ln 5, g = 5 h = ln + 3 3, h 3 + 3 = + 3
Ableitungsregeln: Lösungen L4 a f = ln = ln, f = g = ln 3 = 3 ln, h = ln n = n ln, g = 3 h = n b f = ln +, f = + g = ln +, g = + h = ln + +, h = + + + c f = ln + +, f = + + + + = g = ln, g = +
Ableitungsregeln: Lösungen L5 a f = ln, f = + + + g = ln, g = + + + + + h = ln = ln = ln 4 + h = = ln ln = ln b f = ln = + ln = ln ln + + f = = + + g = ln 3 + +, g = 3 + + + h = ln 5 4, h = 5 c f = ln, f = + g =, g = + + + h =, h = + + + + =
Ableitungsregeln: Lösungen L6 a f = ln cos, f = sin cos, g = ln sin, h = ln tan, g = cos sin, h = + tan tan b f = ln cos 3, f 3 sin 3 = cos 3, g = ln sin 5, g = h = ln tan, = sin cos cos 5 sin 5, h = + tan tan = sin cos c f = ln cos, f = sin cos, g = ln sin 3, g = cos 3 3 sin 3, h = ln tan + 4, h = + 4 cos + 4 sin + 4 sin 3 d f = ln cos 3, f = 3 cos 3, g = ln h = ln sin 3, g = tan + 4, h = 3 cos 3 3 sin 3, + 4 cos + 4 sin + 4.
Logarithmische Differentiation L7 a f = ln cos = ln cos, f = sin cos, g = ln + sin = ln + sin, g = + cos + sin h = ln tan = ln tan, h = sin cos b f = ln 3 cos = 3 ln cos, f = sin 3 cos, g = ln 4 + sin = 4 ln + sin, g = + cos 4 + sin tan tan = 3 h = ln 3 tan = 3 ln tan, h = 3 = cos 3 sin cos = 3 sin cos = 3 sin. tan cos = L8 a f = ln, f = ln ln g = 3, g = 3 ln ln + h =, h = ln 3 ln + ln + ln L9 f = ln + ln, f + + ln = g = ln, g = + ln h = lnsin, h = cos sin = cot L30 7. Logarithmische Differentiation f f = u v, ln f = ln u + ln v, f u f = f u + v u = u v v u + v = u v + v u v = u u + v v f = u v, ln f = ln u ln v, f f u f = f u v = u u v v u v = u v v uv v = u u v v = u v v u v
Logarithmische Differentiation L3 a y = 5, y = 5 ln 5, y = 3 4, y = 4 3 4 ln 3 y = 7 3, y = 3 7 3 ln 7 b y = e, ln y = e ln, y = e e + ln y = e e, ln y = ln + e, y = e e + e c y = e, ln y = ln e = ln y y =, y = e y = e, y = e = e L3 a y = e, ln y = ln, y = e y = 4 3 e, ln y = ln 4 + 3 ln, y = 4 3 e 3 b y = +, ln y = ln + = ln + y = + ln + + + y = + 7, ln y = ln + 7 = ln + 7 y = + 7 ln 4 + 7 + + 7 c y = e 3 4, ln y = ln e 3 4 = + 3 ln + ln 4 ln + + + y y = + 3 + 4 +, y = e 3 4 + + 3 + 4 +
Logarithmische Differentiation L33 a y = + 4 y, ln y = ln + 4 ln +, y = + 4 + y = + 4 4 + = + 3 3 b y =, ln y = ln, y = ln y y = ln c y =, ln y = ln = ln = ln y y = ln +, y = ln + = ln + L34 a y = +, ln y = ln ln +, y y = + y = + = + + b y = +, ln y = ln + ln, y y = + y = + + 4 = c y =, ln y = ln + ln ln + 3 ln + 4 + 3 + 4 y y = + + 3 + 4, y = + 3 + 4 + + 3 + 4 + d y =, ln y = ln + ln, y y = + y = + + = +
Logarithmische Differentiation L35 a y = sin 3, ln y = 3 ln sin y y = 3 ln sin + 3 cos sin, y = sin 3 3 ln sin + 3 cos sin b y = 3 cos, ln y = ln 3 + ln + ln cos y = 3 cos + ln cos tan = 3 cos + ln cos tan c y = tan e, ln y = ln tan y y = cos tan, y = tan e cos tan = e cos tan