Analysis Beispiele

$f (x) = 3 \ln (\sec (x) + \tan (x))$

Schritt 1

Bestimme die erste Ableitung.

Tippen, um mehr Schritte zu sehen ...

Schritt 1.1

Da $3$ konstant bezüglich $x$ ist, ist die Ableitung von $3 \ln (\sec (x) + \tan (x))$ nach $x$ gleich $3 \frac{d}{d x} [\ln (\sec (x) + \tan (x))]$ .

$3 \frac{d}{d x} [\ln (\sec (x) + \tan (x))]$

Schritt 1.2

Differenziere unter Anwendung der Kettenregel, die besagt, dass $\frac{d}{d x} [f (g (x))]$ ist $f' (g (x)) g' (x)$ , mit $f (x) = \ln (x)$ und $g (x) = \sec (x) + \tan (x)$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 1.2.1

Um die Kettenregel anzuwenden, ersetze $u$ durch $\sec (x) + \tan (x)$ .

$3 (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [\sec (x) + \tan (x)])$

Schritt 1.2.2

Die Ableitung von $\ln (u)$ nach $u$ ist $\frac{1}{u}$ .

$3 (\frac{1}{u} \frac{d}{d x} [\sec (x) + \tan (x)])$

Schritt 1.2.3

Ersetze alle $u$ durch $\sec (x) + \tan (x)$ .

$3 (\frac{1}{\sec (x) + \tan (x)} \frac{d}{d x} [\sec (x) + \tan (x)])$

Schritt 1.3

Differenziere unter Anwendung der Summenregel.

Tippen, um mehr Schritte zu sehen ...

Schritt 1.3.1

Kombiniere $\frac{1}{\sec (x) + \tan (x)}$ und $3$ .

$\frac{3}{\sec (x) + \tan (x)} \frac{d}{d x} [\sec (x) + \tan (x)]$

Schritt 1.3.2

Gemäß der Summenregel ist die Ableitung von $\sec (x) + \tan (x)$ nach $x$ $\frac{d}{d x} [\sec (x)] + \frac{d}{d x} [\tan (x)]$ .

$\frac{3}{\sec (x) + \tan (x)} (\frac{d}{d x} [\sec (x)] + \frac{d}{d x} [\tan (x)])$

Schritt 1.4

Die Ableitung von $\sec (x)$ nach $x$ ist $\sec (x) \tan (x)$ .

$\frac{3}{\sec (x) + \tan (x)} (\sec (x) \tan (x) + \frac{d}{d x} [\tan (x)])$

Schritt 1.5

Die Ableitung von $\tan (x)$ nach $x$ ist $\sec^{2} (x)$ .

$\frac{3}{\sec (x) + \tan (x)} (\sec (x) \tan (x) + \sec^{2} (x))$

Schritt 1.6

Vereinfache.

Tippen, um mehr Schritte zu sehen ...

Schritt 1.6.1

Stelle die Faktoren von $\frac{3}{\sec (x) + \tan (x)} (\sec (x) \tan (x) + \sec^{2} (x))$ um.

$(\sec (x) \tan (x) + \sec^{2} (x)) \frac{3}{\sec (x) + \tan (x)}$

Schritt 1.6.2

Wende das Distributivgesetz an.

$\sec (x) \tan (x) \frac{3}{\sec (x) + \tan (x)} + \sec^{2} (x) \frac{3}{\sec (x) + \tan (x)}$

Schritt 1.6.3

Multipliziere $\sec (x) \tan (x) \frac{3}{\sec (x) + \tan (x)}$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 1.6.3.1

Kombiniere $\frac{3}{\sec (x) + \tan (x)}$ und $\sec (x)$ .

$\frac{3 \sec (x)}{\sec (x) + \tan (x)} \tan (x) + \sec^{2} (x) \frac{3}{\sec (x) + \tan (x)}$

Schritt 1.6.3.2

Kombiniere $\frac{3 \sec (x)}{\sec (x) + \tan (x)}$ und $\tan (x)$ .

$\frac{3 \sec (x) \tan (x)}{\sec (x) + \tan (x)} + \sec^{2} (x) \frac{3}{\sec (x) + \tan (x)}$

Schritt 1.6.4

Kombiniere $\sec^{2} (x)$ und $\frac{3}{\sec (x) + \tan (x)}$ .

$\frac{3 \sec (x) \tan (x)}{\sec (x) + \tan (x)} + \frac{\sec^{2} (x) \cdot 3}{\sec (x) + \tan (x)}$

Schritt 1.6.5

Bringe $3$ auf die linke Seite von $\sec^{2} (x)$ .

$\frac{3 \sec (x) \tan (x)}{\sec (x) + \tan (x)} + \frac{3 \sec^{2} (x)}{\sec (x) + \tan (x)}$

Schritt 1.6.6

Vereinige die Zähler über dem gemeinsamen Nenner.

$\frac{3 \sec (x) \tan (x) + 3 \sec^{2} (x)}{\sec (x) + \tan (x)}$

Schritt 1.6.7

Faktorisiere $3 \sec (x)$ aus $3 \sec (x) \tan (x) + 3 \sec^{2} (x)$ heraus.

Tippen, um mehr Schritte zu sehen ...

Schritt 1.6.7.1

Faktorisiere $3 \sec (x)$ aus $3 \sec (x) \tan (x)$ heraus.

$\frac{3 \sec (x) (\tan (x)) + 3 \sec^{2} (x)}{\sec (x) + \tan (x)}$

Schritt 1.6.7.2

Faktorisiere $3 \sec (x)$ aus $3 \sec^{2} (x)$ heraus.

$\frac{3 \sec (x) (\tan (x)) + 3 \sec (x) (\sec (x))}{\sec (x) + \tan (x)}$

Schritt 1.6.7.3

Faktorisiere $3 \sec (x)$ aus $3 \sec (x) (\tan (x)) + 3 \sec (x) (\sec (x))$ heraus.

$f' (x) = \frac{3 \sec (x) (\tan (x) + \sec (x))}{\sec (x) + \tan (x)}$

Schritt 2

Bestimme die zweite Ableitung.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.1

Da $3$ konstant bezüglich $x$ ist, ist die Ableitung von $\frac{3 \sec (x) (\tan (x) + \sec (x))}{\sec (x) + \tan (x)}$ nach $x$ gleich $3 \frac{d}{d x} [\frac{\sec (x) (\tan (x) + \sec (x))}{\sec (x) + \tan (x)}]$ .

$3 \frac{d}{d x} [\frac{\sec (x) (\tan (x) + \sec (x))}{\sec (x) + \tan (x)}]$

Schritt 2.2

Differenziere unter Anwendung der Quotientenregel, die besagt, dass $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ gleich $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ ist mit $f (x) = \sec (x) (\tan (x) + \sec (x))$ und $g (x) = \sec (x) + \tan (x)$ .

$3 \frac{(\sec (x) + \tan (x)) \frac{d}{d x} [\sec (x) (\tan (x) + \sec (x))] - \sec (x) (\tan (x) + \sec (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.3

Differenziere unter Anwendung der Produktregel, die besagt, dass $\frac{d}{d x} [f (x) g (x)]$ gleich $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ ist mit $f (x) = \sec (x)$ und $g (x) = \tan (x) + \sec (x)$ .

$3 \frac{(\sec (x) + \tan (x)) (\sec (x) \frac{d}{d x} [\tan (x) + \sec (x)] + (\tan (x) + \sec (x)) \frac{d}{d x} [\sec (x)]) - \sec (x) (\tan (x) + \sec (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.4

Gemäß der Summenregel ist die Ableitung von $\tan (x) + \sec (x)$ nach $x$ $\frac{d}{d x} [\tan (x)] + \frac{d}{d x} [\sec (x)]$ .

$3 \frac{(\sec (x) + \tan (x)) (\sec (x) (\frac{d}{d x} [\tan (x)] + \frac{d}{d x} [\sec (x)]) + (\tan (x) + \sec (x)) \frac{d}{d x} [\sec (x)]) - \sec (x) (\tan (x) + \sec (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.5

Die Ableitung von $\tan (x)$ nach $x$ ist $\sec^{2} (x)$ .

$3 \frac{(\sec (x) + \tan (x)) (\sec (x) (\sec^{2} (x) + \frac{d}{d x} [\sec (x)]) + (\tan (x) + \sec (x)) \frac{d}{d x} [\sec (x)]) - \sec (x) (\tan (x) + \sec (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.6

Die Ableitung von $\sec (x)$ nach $x$ ist $\sec (x) \tan (x)$ .

$3 \frac{(\sec (x) + \tan (x)) (\sec (x) (\sec^{2} (x) + \sec (x) \tan (x)) + (\tan (x) + \sec (x)) \frac{d}{d x} [\sec (x)]) - \sec (x) (\tan (x) + \sec (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.7

Die Ableitung von $\sec (x)$ nach $x$ ist $\sec (x) \tan (x)$ .

$3 \frac{(\sec (x) + \tan (x)) (\sec (x) (\sec^{2} (x) + \sec (x) \tan (x)) + (\tan (x) + \sec (x)) (\sec (x) \tan (x))) - \sec (x) (\tan (x) + \sec (x)) \frac{d}{d x} [\sec (x) + \tan (x)]}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.8

Gemäß der Summenregel ist die Ableitung von $\sec (x) + \tan (x)$ nach $x$ $\frac{d}{d x} [\sec (x)] + \frac{d}{d x} [\tan (x)]$ .

$3 \frac{(\sec (x) + \tan (x)) (\sec (x) (\sec^{2} (x) + \sec (x) \tan (x)) + (\tan (x) + \sec (x)) \sec (x) \tan (x)) - \sec (x) (\tan (x) + \sec (x)) (\frac{d}{d x} [\sec (x)] + \frac{d}{d x} [\tan (x)])}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.9

Die Ableitung von $\sec (x)$ nach $x$ ist $\sec (x) \tan (x)$ .

$3 \frac{(\sec (x) + \tan (x)) (\sec (x) (\sec^{2} (x) + \sec (x) \tan (x)) + (\tan (x) + \sec (x)) \sec (x) \tan (x)) - \sec (x) (\tan (x) + \sec (x)) (\sec (x) \tan (x) + \frac{d}{d x} [\tan (x)])}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.10

Die Ableitung von $\tan (x)$ nach $x$ ist $\sec^{2} (x)$ .

$3 \frac{(\sec (x) + \tan (x)) (\sec (x) (\sec^{2} (x) + \sec (x) \tan (x)) + (\tan (x) + \sec (x)) \sec (x) \tan (x)) - \sec (x) (\tan (x) + \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.11

Kombiniere $3$ und $\frac{(\sec (x) + \tan (x)) (\sec (x) (\sec^{2} (x) + \sec (x) \tan (x)) + (\tan (x) + \sec (x)) \sec (x) \tan (x)) - \sec (x) (\tan (x) + \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$ .

$\frac{3 ((\sec (x) + \tan (x)) (\sec (x) (\sec^{2} (x) + \sec (x) \tan (x)) + (\tan (x) + \sec (x)) \sec (x) \tan (x)) - \sec (x) (\tan (x) + \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12

Vereinfache.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.12.1

Wende das Distributivgesetz an.

$\frac{3 ((\sec (x) + \tan (x)) (\sec (x) \sec^{2} (x) + \sec (x) (\sec (x) \tan (x)) + (\tan (x) + \sec (x)) \sec (x) \tan (x)) - \sec (x) (\tan (x) + \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.2

Wende das Distributivgesetz an.

$\frac{3 ((\sec (x) + \tan (x)) (\sec (x) \sec^{2} (x) + \sec (x) (\sec (x) \tan (x)) + (\tan (x) \sec (x) + \sec (x) \sec (x)) \tan (x)) - \sec (x) (\tan (x) + \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.3

Wende das Distributivgesetz an.

$\frac{3 ((\sec (x) + \tan (x)) (\sec (x) \sec^{2} (x) + \sec (x) (\sec (x) \tan (x)) + \tan (x) \sec (x) \tan (x) + \sec (x) \sec (x) \tan (x)) - \sec (x) (\tan (x) + \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.4

Wende das Distributivgesetz an.

$\frac{3 ((\sec (x) + \tan (x)) (\sec (x) \sec^{2} (x) + \sec (x) (\sec (x) \tan (x)) + \tan (x) \sec (x) \tan (x) + \sec (x) \sec (x) \tan (x)) + (- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.5

Wende das Distributivgesetz an.

$\frac{3 ((\sec (x) + \tan (x)) (\sec (x) \sec^{2} (x) + \sec (x) (\sec (x) \tan (x)) + \tan (x) \sec (x) \tan (x) + \sec (x) \sec (x) \tan (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6

Vereinfache den Zähler.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.12.6.1

Vereinfache jeden Term.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.12.6.1.1

Vereinfache jeden Term.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.12.6.1.1.1

Multipliziere $\sec (x)$ mit $\sec^{2} (x)$ durch Addieren der Exponenten.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.12.6.1.1.1.1

Mutltipliziere $\sec (x)$ mit $\sec^{2} (x)$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 2.12.6.1.1.1.1.1

Potenziere $\sec (x)$ mit $1$ .

$\frac{3 ((\sec (x) + \tan (x)) (\sec^{1} (x) \sec^{2} (x) + \sec (x) (\sec (x) \tan (x)) + \tan (x) \sec (x) \tan (x) + \sec (x) \sec (x) \tan (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.1.1.1.2

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$\frac{3 ((\sec (x) + \tan (x)) ({\sec (x)}^{1 + 2} + \sec (x) (\sec (x) \tan (x)) + \tan (x) \sec (x) \tan (x) + \sec (x) \sec (x) \tan (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.1.1.2

Addiere $1$ und $2$ .

$\frac{3 ((\sec (x) + \tan (x)) (\sec^{3} (x) + \sec (x) (\sec (x) \tan (x)) + \tan (x) \sec (x) \tan (x) + \sec (x) \sec (x) \tan (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.1.2

Multipliziere $\sec (x) (\sec (x) \tan (x))$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 2.12.6.1.1.2.1

Potenziere $\sec (x)$ mit $1$ .

$\frac{3 ((\sec (x) + \tan (x)) (\sec^{3} (x) + \sec^{1} (x) \sec (x) \tan (x) + \tan (x) \sec (x) \tan (x) + \sec (x) \sec (x) \tan (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.1.2.2

Potenziere $\sec (x)$ mit $1$ .

$\frac{3 ((\sec (x) + \tan (x)) (\sec^{3} (x) + \sec^{1} (x) \sec^{1} (x) \tan (x) + \tan (x) \sec (x) \tan (x) + \sec (x) \sec (x) \tan (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.1.2.3

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$\frac{3 ((\sec (x) + \tan (x)) (\sec^{3} (x) + {\sec (x)}^{1 + 1} \tan (x) + \tan (x) \sec (x) \tan (x) + \sec (x) \sec (x) \tan (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.1.2.4

Addiere $1$ und $1$ .

$\frac{3 ((\sec (x) + \tan (x)) (\sec^{3} (x) + \sec^{2} (x) \tan (x) + \tan (x) \sec (x) \tan (x) + \sec (x) \sec (x) \tan (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.1.3

Multipliziere $\tan (x) \sec (x) \tan (x)$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 2.12.6.1.1.3.1

Potenziere $\tan (x)$ mit $1$ .

$\frac{3 ((\sec (x) + \tan (x)) (\sec^{3} (x) + \sec^{2} (x) \tan (x) + \tan^{1} (x) \tan (x) \sec (x) + \sec (x) \sec (x) \tan (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.1.3.2

Potenziere $\tan (x)$ mit $1$ .

$\frac{3 ((\sec (x) + \tan (x)) (\sec^{3} (x) + \sec^{2} (x) \tan (x) + \tan^{1} (x) \tan^{1} (x) \sec (x) + \sec (x) \sec (x) \tan (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.1.3.3

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$\frac{3 ((\sec (x) + \tan (x)) (\sec^{3} (x) + \sec^{2} (x) \tan (x) + {\tan (x)}^{1 + 1} \sec (x) + \sec (x) \sec (x) \tan (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.1.3.4

Addiere $1$ und $1$ .

$\frac{3 ((\sec (x) + \tan (x)) (\sec^{3} (x) + \sec^{2} (x) \tan (x) + \tan^{2} (x) \sec (x) + \sec (x) \sec (x) \tan (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.1.4

Multipliziere $\sec (x) \sec (x)$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 2.12.6.1.1.4.1

Potenziere $\sec (x)$ mit $1$ .

$\frac{3 ((\sec (x) + \tan (x)) (\sec^{3} (x) + \sec^{2} (x) \tan (x) + \tan^{2} (x) \sec (x) + \sec^{1} (x) \sec (x) \tan (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.1.4.2

Potenziere $\sec (x)$ mit $1$ .

$\frac{3 ((\sec (x) + \tan (x)) (\sec^{3} (x) + \sec^{2} (x) \tan (x) + \tan^{2} (x) \sec (x) + \sec^{1} (x) \sec^{1} (x) \tan (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.1.4.3

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$\frac{3 ((\sec (x) + \tan (x)) (\sec^{3} (x) + \sec^{2} (x) \tan (x) + \tan^{2} (x) \sec (x) + {\sec (x)}^{1 + 1} \tan (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.1.4.4

Addiere $1$ und $1$ .

$\frac{3 ((\sec (x) + \tan (x)) (\sec^{3} (x) + \sec^{2} (x) \tan (x) + \tan^{2} (x) \sec (x) + \sec^{2} (x) \tan (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.2

Addiere $\sec^{2} (x) \tan (x)$ und $\sec^{2} (x) \tan (x)$ .

$\frac{3 ((\sec (x) + \tan (x)) (\sec^{3} (x) + 2 \sec^{2} (x) \tan (x) + \tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.3

Multipliziere $(\sec (x) + \tan (x)) (\sec^{3} (x) + 2 \sec^{2} (x) \tan (x) + \tan^{2} (x) \sec (x))$ aus durch Multiplizieren jedes Terms des ersten Ausdrucks mit jedem Term des zweiten Ausdrucks.

$\frac{3 (\sec (x) \sec^{3} (x) + \sec (x) (2 \sec^{2} (x) \tan (x)) + \sec (x) (\tan^{2} (x) \sec (x)) + \tan (x) \sec^{3} (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.4

Vereinfache jeden Term.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.12.6.1.4.1

Multipliziere $\sec (x)$ mit $\sec^{3} (x)$ durch Addieren der Exponenten.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.12.6.1.4.1.1

Mutltipliziere $\sec (x)$ mit $\sec^{3} (x)$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 2.12.6.1.4.1.1.1

Potenziere $\sec (x)$ mit $1$ .

$\frac{3 (\sec^{1} (x) \sec^{3} (x) + \sec (x) (2 \sec^{2} (x) \tan (x)) + \sec (x) (\tan^{2} (x) \sec (x)) + \tan (x) \sec^{3} (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.4.1.1.2

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$\frac{3 ({\sec (x)}^{1 + 3} + \sec (x) (2 \sec^{2} (x) \tan (x)) + \sec (x) (\tan^{2} (x) \sec (x)) + \tan (x) \sec^{3} (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.4.1.2

Addiere $1$ und $3$ .

$\frac{3 (\sec^{4} (x) + \sec (x) (2 \sec^{2} (x) \tan (x)) + \sec (x) (\tan^{2} (x) \sec (x)) + \tan (x) \sec^{3} (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.4.2

Schreibe neu unter Anwendung des Kommutativgesetzes der Multiplikation.

$\frac{3 (\sec^{4} (x) + 2 \sec (x) \sec^{2} (x) \tan (x) + \sec (x) (\tan^{2} (x) \sec (x)) + \tan (x) \sec^{3} (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.4.3

Multipliziere $\sec (x)$ mit $\sec^{2} (x)$ durch Addieren der Exponenten.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.12.6.1.4.3.1

Bewege $\sec^{2} (x)$ .

$\frac{3 (\sec^{4} (x) + 2 (\sec^{2} (x) \sec (x)) \tan (x) + \sec (x) (\tan^{2} (x) \sec (x)) + \tan (x) \sec^{3} (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.4.3.2

Mutltipliziere $\sec^{2} (x)$ mit $\sec (x)$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 2.12.6.1.4.3.2.1

Potenziere $\sec (x)$ mit $1$ .

$\frac{3 (\sec^{4} (x) + 2 (\sec^{2} (x) \sec^{1} (x)) \tan (x) + \sec (x) (\tan^{2} (x) \sec (x)) + \tan (x) \sec^{3} (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.4.3.2.2

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$\frac{3 (\sec^{4} (x) + 2 {\sec (x)}^{2 + 1} \tan (x) + \sec (x) (\tan^{2} (x) \sec (x)) + \tan (x) \sec^{3} (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.4.3.3

Addiere $2$ und $1$ .

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec (x) (\tan^{2} (x) \sec (x)) + \tan (x) \sec^{3} (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.4.4

Multipliziere $\sec (x) (\tan^{2} (x) \sec (x))$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 2.12.6.1.4.4.1

Potenziere $\sec (x)$ mit $1$ .

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{1} (x) \sec (x) \tan^{2} (x) + \tan (x) \sec^{3} (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.4.4.2

Potenziere $\sec (x)$ mit $1$ .

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{1} (x) \sec^{1} (x) \tan^{2} (x) + \tan (x) \sec^{3} (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.4.4.3

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + {\sec (x)}^{1 + 1} \tan^{2} (x) + \tan (x) \sec^{3} (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.4.4.4

Addiere $1$ und $1$ .

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + \tan (x) \sec^{3} (x) + \tan (x) (2 \sec^{2} (x) \tan (x)) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.4.5

Schreibe neu unter Anwendung des Kommutativgesetzes der Multiplikation.

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + \tan (x) \sec^{3} (x) + 2 \tan (x) \sec^{2} (x) \tan (x) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.4.6

Multipliziere $2 \tan (x) \sec^{2} (x) \tan (x)$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 2.12.6.1.4.6.1

Potenziere $\tan (x)$ mit $1$ .

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + \tan (x) \sec^{3} (x) + 2 (\tan^{1} (x) \tan (x)) \sec^{2} (x) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.4.6.2

Potenziere $\tan (x)$ mit $1$ .

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + \tan (x) \sec^{3} (x) + 2 (\tan^{1} (x) \tan^{1} (x)) \sec^{2} (x) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.4.6.3

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + \tan (x) \sec^{3} (x) + 2 {\tan (x)}^{1 + 1} \sec^{2} (x) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.4.6.4

Addiere $1$ und $1$ .

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + \tan (x) \sec^{3} (x) + 2 \tan^{2} (x) \sec^{2} (x) + \tan (x) (\tan^{2} (x) \sec (x))) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.4.7

Multipliziere $\tan (x)$ mit $\tan^{2} (x)$ durch Addieren der Exponenten.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.12.6.1.4.7.1

Bewege $\tan^{2} (x)$ .

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + \tan (x) \sec^{3} (x) + 2 \tan^{2} (x) \sec^{2} (x) + \tan^{2} (x) \tan (x) \sec (x)) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.4.7.2

Mutltipliziere $\tan^{2} (x)$ mit $\tan (x)$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 2.12.6.1.4.7.2.1

Potenziere $\tan (x)$ mit $1$ .

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + \tan (x) \sec^{3} (x) + 2 \tan^{2} (x) \sec^{2} (x) + \tan^{2} (x) \tan^{1} (x) \sec (x)) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.4.7.2.2

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + \tan (x) \sec^{3} (x) + 2 \tan^{2} (x) \sec^{2} (x) + {\tan (x)}^{2 + 1} \sec (x)) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.4.7.3

Addiere $2$ und $1$ .

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + \tan (x) \sec^{3} (x) + 2 \tan^{2} (x) \sec^{2} (x) + \tan^{3} (x) \sec (x)) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.5

Stelle die Faktoren von $\tan (x) \sec^{3} (x)$ um.

$\frac{3 (\sec^{4} (x) + 2 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + \sec^{3} (x) \tan (x) + 2 \tan^{2} (x) \sec^{2} (x) + \tan^{3} (x) \sec (x)) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.6

Addiere $2 \sec^{3} (x) \tan (x)$ und $\sec^{3} (x) \tan (x)$ .

$\frac{3 (\sec^{4} (x) + 3 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + 2 \tan^{2} (x) \sec^{2} (x) + \tan^{3} (x) \sec (x)) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.7

Stelle die Faktoren von $2 \tan^{2} (x) \sec^{2} (x)$ um.

$\frac{3 (\sec^{4} (x) + 3 \sec^{3} (x) \tan (x) + \sec^{2} (x) \tan^{2} (x) + 2 \sec^{2} (x) \tan^{2} (x) + \tan^{3} (x) \sec (x)) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.8

Addiere $\sec^{2} (x) \tan^{2} (x)$ und $2 \sec^{2} (x) \tan^{2} (x)$ .

$\frac{3 (\sec^{4} (x) + 3 \sec^{3} (x) \tan (x) + 3 \sec^{2} (x) \tan^{2} (x) + \tan^{3} (x) \sec (x)) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.9

Wende das Distributivgesetz an.

$\frac{3 \sec^{4} (x) + 3 (3 \sec^{3} (x) \tan (x)) + 3 (3 \sec^{2} (x) \tan^{2} (x)) + 3 (\tan^{3} (x) \sec (x)) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.10

Vereinfache.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.12.6.1.10.1

Mutltipliziere $3$ mit $3$ .

$\frac{3 \sec^{4} (x) + 9 (\sec^{3} (x) \tan (x)) + 3 (3 \sec^{2} (x) \tan^{2} (x)) + 3 (\tan^{3} (x) \sec (x)) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.10.2

Mutltipliziere $3$ mit $3$ .

$\frac{3 \sec^{4} (x) + 9 (\sec^{3} (x) \tan (x)) + 9 (\sec^{2} (x) \tan^{2} (x)) + 3 \tan^{3} (x) \sec (x) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.11

Entferne die Klammern.

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 ((- \sec (x) \tan (x) - \sec (x) \sec (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.12

Multipliziere $- \sec (x) \sec (x)$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 2.12.6.1.12.1

Potenziere $\sec (x)$ mit $1$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 ((- \sec (x) \tan (x) - (\sec^{1} (x) \sec (x))) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.12.2

Potenziere $\sec (x)$ mit $1$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 ((- \sec (x) \tan (x) - (\sec^{1} (x) \sec^{1} (x))) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.12.3

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 ((- \sec (x) \tan (x) - {\sec (x)}^{1 + 1}) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.12.4

Addiere $1$ und $1$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 ((- \sec (x) \tan (x) - \sec^{2} (x)) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.13

Multipliziere $(- \sec (x) \tan (x) - \sec^{2} (x)) (\sec (x) \tan (x) + \sec^{2} (x))$ aus unter Verwendung der FOIL-Methode.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.12.6.1.13.1

Wende das Distributivgesetz an.

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec (x) \tan (x) (\sec (x) \tan (x) + \sec^{2} (x)) - \sec^{2} (x) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.13.2

Wende das Distributivgesetz an.

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec (x) \tan (x) (\sec (x) \tan (x)) - \sec (x) \tan (x) \sec^{2} (x) - \sec^{2} (x) (\sec (x) \tan (x) + \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.13.3

Wende das Distributivgesetz an.

Schritt 2.12.6.1.14

Vereinfache und fasse gleichartige Terme zusammen.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.12.6.1.14.1

Vereinfache jeden Term.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.12.6.1.14.1.1

Multipliziere $- \sec (x) \tan (x) (\sec (x) \tan (x))$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 2.12.6.1.14.1.1.1

Potenziere $\sec (x)$ mit $1$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- (\sec^{1} (x) \sec (x)) \tan (x) \tan (x) - \sec (x) \tan (x) \sec^{2} (x) - \sec^{2} (x) (\sec (x) \tan (x)) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.14.1.1.2

Potenziere $\sec (x)$ mit $1$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- (\sec^{1} (x) \sec^{1} (x)) \tan (x) \tan (x) - \sec (x) \tan (x) \sec^{2} (x) - \sec^{2} (x) (\sec (x) \tan (x)) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.14.1.1.3

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- {\sec (x)}^{1 + 1} \tan (x) \tan (x) - \sec (x) \tan (x) \sec^{2} (x) - \sec^{2} (x) (\sec (x) \tan (x)) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.14.1.1.4

Addiere $1$ und $1$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan (x) \tan (x) - \sec (x) \tan (x) \sec^{2} (x) - \sec^{2} (x) (\sec (x) \tan (x)) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.14.1.1.5

Potenziere $\tan (x)$ mit $1$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) (\tan^{1} (x) \tan (x)) - \sec (x) \tan (x) \sec^{2} (x) - \sec^{2} (x) (\sec (x) \tan (x)) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.14.1.1.6

Potenziere $\tan (x)$ mit $1$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) (\tan^{1} (x) \tan^{1} (x)) - \sec (x) \tan (x) \sec^{2} (x) - \sec^{2} (x) (\sec (x) \tan (x)) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.14.1.1.7

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) {\tan (x)}^{1 + 1} - \sec (x) \tan (x) \sec^{2} (x) - \sec^{2} (x) (\sec (x) \tan (x)) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.14.1.1.8

Addiere $1$ und $1$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x) - \sec (x) \tan (x) \sec^{2} (x) - \sec^{2} (x) (\sec (x) \tan (x)) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.14.1.2

Multipliziere $\sec (x)$ mit $\sec^{2} (x)$ durch Addieren der Exponenten.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.12.6.1.14.1.2.1

Bewege $\sec^{2} (x)$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x) - (\sec^{2} (x) \sec (x)) \tan (x) - \sec^{2} (x) (\sec (x) \tan (x)) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.14.1.2.2

Mutltipliziere $\sec^{2} (x)$ mit $\sec (x)$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 2.12.6.1.14.1.2.2.1

Potenziere $\sec (x)$ mit $1$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x) - (\sec^{2} (x) \sec^{1} (x)) \tan (x) - \sec^{2} (x) (\sec (x) \tan (x)) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.14.1.2.2.2

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x) - {\sec (x)}^{2 + 1} \tan (x) - \sec^{2} (x) (\sec (x) \tan (x)) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.14.1.2.3

Addiere $2$ und $1$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x) - \sec^{3} (x) \tan (x) - \sec^{2} (x) (\sec (x) \tan (x)) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.14.1.3

Multipliziere $\sec^{2} (x)$ mit $\sec (x)$ durch Addieren der Exponenten.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.12.6.1.14.1.3.1

Bewege $\sec (x)$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x) - \sec^{3} (x) \tan (x) - (\sec (x) \sec^{2} (x)) \tan (x) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.14.1.3.2

Mutltipliziere $\sec (x)$ mit $\sec^{2} (x)$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 2.12.6.1.14.1.3.2.1

Potenziere $\sec (x)$ mit $1$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x) - \sec^{3} (x) \tan (x) - (\sec^{1} (x) \sec^{2} (x)) \tan (x) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.14.1.3.2.2

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x) - \sec^{3} (x) \tan (x) - {\sec (x)}^{1 + 2} \tan (x) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.14.1.3.3

Addiere $1$ und $2$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x) - \sec^{3} (x) \tan (x) - \sec^{3} (x) \tan (x) - \sec^{2} (x) \sec^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.14.1.4

Multipliziere $\sec^{2} (x)$ mit $\sec^{2} (x)$ durch Addieren der Exponenten.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.12.6.1.14.1.4.1

Bewege $\sec^{2} (x)$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x) - \sec^{3} (x) \tan (x) - \sec^{3} (x) \tan (x) - (\sec^{2} (x) \sec^{2} (x)))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.14.1.4.2

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x) - \sec^{3} (x) \tan (x) - \sec^{3} (x) \tan (x) - {\sec (x)}^{2 + 2})}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.14.1.4.3

Addiere $2$ und $2$ .

Schritt 2.12.6.1.14.2

Subtrahiere $\sec^{3} (x) \tan (x)$ von $- \sec^{3} (x) \tan (x)$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x) - 2 \sec^{3} (x) \tan (x) - \sec^{4} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.15

Wende das Distributivgesetz an.

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) + 3 (- \sec^{2} (x) \tan^{2} (x)) + 3 (- 2 \sec^{3} (x) \tan (x)) + 3 (- \sec^{4} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.16

Vereinfache.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.12.6.1.16.1

Mutltipliziere $- 1$ mit $3$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) - 3 (\sec^{2} (x) \tan^{2} (x)) + 3 (- 2 \sec^{3} (x) \tan (x)) + 3 (- \sec^{4} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.16.2

Mutltipliziere $- 2$ mit $3$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) - 3 (\sec^{2} (x) \tan^{2} (x)) - 6 (\sec^{3} (x) \tan (x)) + 3 (- \sec^{4} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.16.3

Mutltipliziere $- 1$ mit $3$ .

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) - 3 (\sec^{2} (x) \tan^{2} (x)) - 6 (\sec^{3} (x) \tan (x)) - 3 \sec^{4} (x)}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.1.17

Entferne die Klammern.

$\frac{3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) - 3 \sec^{2} (x) \tan^{2} (x) - 6 \sec^{3} (x) \tan (x) - 3 \sec^{4} (x)}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.2

Vereine die Terme mit entgegengesetztem Vorzeichen in $3 \sec^{4} (x) + 9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) - 3 \sec^{2} (x) \tan^{2} (x) - 6 \sec^{3} (x) \tan (x) - 3 \sec^{4} (x)$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 2.12.6.2.1

Subtrahiere $3 \sec^{4} (x)$ von $3 \sec^{4} (x)$ .

$\frac{9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) - 3 \sec^{2} (x) \tan^{2} (x) - 6 \sec^{3} (x) \tan (x) + 0}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.2.2

Addiere $9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) - 3 \sec^{2} (x) \tan^{2} (x) - 6 \sec^{3} (x) \tan (x)$ und $0$ .

$\frac{9 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) - 3 \sec^{2} (x) \tan^{2} (x) - 6 \sec^{3} (x) \tan (x)}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.3

Subtrahiere $6 \sec^{3} (x) \tan (x)$ von $9 \sec^{3} (x) \tan (x)$ .

$\frac{3 \sec^{3} (x) \tan (x) + 9 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x) - 3 \sec^{2} (x) \tan^{2} (x)}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.6.4

Subtrahiere $3 \sec^{2} (x) \tan^{2} (x)$ von $9 \sec^{2} (x) \tan^{2} (x)$ .

$\frac{3 \sec^{3} (x) \tan (x) + 6 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x)}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.7

Vereinfache den Zähler.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.12.7.1

Faktorisiere $3 \sec (x) \tan (x)$ aus $3 \sec^{3} (x) \tan (x) + 6 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x)$ heraus.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.12.7.1.1

Faktorisiere $3 \sec (x) \tan (x)$ aus $3 \sec^{3} (x) \tan (x)$ heraus.

$\frac{3 \sec (x) \tan (x) (\sec^{2} (x)) + 6 \sec^{2} (x) \tan^{2} (x) + 3 \tan^{3} (x) \sec (x)}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.7.1.2

Faktorisiere $3 \sec (x) \tan (x)$ aus $6 \sec^{2} (x) \tan^{2} (x)$ heraus.

$\frac{3 \sec (x) \tan (x) (\sec^{2} (x)) + 3 \sec (x) \tan (x) (2 \sec (x) \tan (x)) + 3 \tan^{3} (x) \sec (x)}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.7.1.3

Faktorisiere $3 \sec (x) \tan (x)$ aus $3 \tan^{3} (x) \sec (x)$ heraus.

$\frac{3 \sec (x) \tan (x) (\sec^{2} (x)) + 3 \sec (x) \tan (x) (2 \sec (x) \tan (x)) + 3 \sec (x) \tan (x) (\tan^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.7.1.4

Faktorisiere $3 \sec (x) \tan (x)$ aus $3 \sec (x) \tan (x) (\sec^{2} (x)) + 3 \sec (x) \tan (x) (2 \sec (x) \tan (x))$ heraus.

$\frac{3 \sec (x) \tan (x) (\sec^{2} (x) + 2 \sec (x) \tan (x)) + 3 \sec (x) \tan (x) (\tan^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.7.1.5

Faktorisiere $3 \sec (x) \tan (x)$ aus $3 \sec (x) \tan (x) (\sec^{2} (x) + 2 \sec (x) \tan (x)) + 3 \sec (x) \tan (x) (\tan^{2} (x))$ heraus.

$\frac{3 \sec (x) \tan (x) (\sec^{2} (x) + 2 \sec (x) \tan (x) + \tan^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.7.2

Faktorisiere unter Verwendung der binomischen Formeln.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.12.7.2.1

Überprüfe, ob der mittlere Term das Zweifache des Produkts der Zahlen ist, die im ersten Term und im dritten Term quadriert werden.

$2 \sec (x) \tan (x) = 2 \cdot \sec (x) \cdot \tan (x)$

Schritt 2.12.7.2.2

Schreibe das Polynom neu.

$\frac{3 \sec (x) \tan (x) (\sec^{2} (x) + 2 \cdot \sec (x) \cdot \tan (x) + \tan^{2} (x))}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.7.2.3

Faktorisiere mithilfe der trinomischen Formel für das perfekte Quadrat $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , wobei $a = \sec (x)$ und $b = \tan (x)$ .

$\frac{3 \sec (x) \tan (x) {(\sec (x) + \tan (x))}^{2}}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.8

Kürze den gemeinsamen Faktor von ${(\sec (x) + \tan (x))}^{2}$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 2.12.8.1

Kürze den gemeinsamen Faktor.

$\frac{3 \sec (x) \tan (x) {(\sec (x) + \tan (x))}^{2}}{{(\sec (x) + \tan (x))}^{2}}$

Schritt 2.12.8.2

Dividiere $3 \sec (x) \tan (x)$ durch $1$ .

$f'' (x) = 3 \sec (x) \tan (x)$

Schritt 3

Die zweite Ableitung von $f (x)$ nach $x$ ist $3 \sec (x) \tan (x)$ .

$3 \sec (x) \tan (x)$