Analysis Beispiele

f (x) = \sin (x) \csc (x)

$f (x) = \sin (x) \csc (x)$

Schritt 1

Differenziere unter Anwendung der Produktregel, die besagt, dass

\frac{d}{d x} [f (x) g (x)]

$\frac{d}{d x} [f (x) g (x)]$ gleich

f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]

$f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ ist mit

f (x) = \sin (x)

$f (x) = \sin (x)$ und

g (x) = \csc (x)

$g (x) = \csc (x)$ .

\sin (x) \frac{d}{d x} [\csc (x)] + \csc (x) \frac{d}{d x} [\sin (x)]

$\sin (x) \frac{d}{d x} [\csc (x)] + \csc (x) \frac{d}{d x} [\sin (x)]$

Schritt 2

Die Ableitung von

\csc (x)

$\csc (x)$ nach

x

$x$ ist

- \csc (x) \cot (x)

$- \csc (x) \cot (x)$ .

\sin (x) (- \csc (x) \cot (x)) + \csc (x) \frac{d}{d x} [\sin (x)]

$\sin (x) (- \csc (x) \cot (x)) + \csc (x) \frac{d}{d x} [\sin (x)]$

Schritt 3

Die Ableitung von

\sin (x)

$\sin (x)$ nach

x

$x$ ist

\cos (x)

$\cos (x)$ .

\sin (x) (- \csc (x) \cot (x)) + \csc (x) \cos (x)

$\sin (x) (- \csc (x) \cot (x)) + \csc (x) \cos (x)$

Schritt 4

Vereinfache.

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Schritt 4.1

Stelle die Terme um.

- \cot (x) \csc (x) \sin (x) + \cos (x) \csc (x)

$- \cot (x) \csc (x) \sin (x) + \cos (x) \csc (x)$

Schritt 4.2

Vereinfache jeden Term.

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Schritt 4.2.1

Schreibe

\cot (x)

$\cot (x)$ mithilfe von Sinus und Kosinus um.

- \frac{\cos (x)}{\sin (x)} \csc (x) \sin (x) + \cos (x) \csc (x)

$- \frac{\cos (x)}{\sin (x)} \csc (x) \sin (x) + \cos (x) \csc (x)$

Schritt 4.2.2

Schreibe

\csc (x)

$\csc (x)$ mithilfe von Sinus und Kosinus um.

- \frac{\cos (x)}{\sin (x)} \cdot \frac{1}{\sin (x)} \sin (x) + \cos (x) \csc (x)

$- \frac{\cos (x)}{\sin (x)} \cdot \frac{1}{\sin (x)} \sin (x) + \cos (x) \csc (x)$

Schritt 4.2.3

Multipliziere

- \frac{\cos (x)}{\sin (x)} \cdot \frac{1}{\sin (x)}

$- \frac{\cos (x)}{\sin (x)} \cdot \frac{1}{\sin (x)}$ .

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Schritt 4.2.3.1

Mutltipliziere

\frac{1}{\sin (x)}

$\frac{1}{\sin (x)}$ mit

\frac{\cos (x)}{\sin (x)}

$\frac{\cos (x)}{\sin (x)}$ .

- \frac{\cos (x)}{\sin (x) \sin (x)} \sin (x) + \cos (x) \csc (x)

$- \frac{\cos (x)}{\sin (x) \sin (x)} \sin (x) + \cos (x) \csc (x)$

Schritt 4.2.3.2

Potenziere

\sin (x)

$\sin (x)$ mit

1

$1$ .

- \frac{\cos (x)}{\sin^{1} (x) \sin (x)} \sin (x) + \cos (x) \csc (x)

$- \frac{\cos (x)}{\sin^{1} (x) \sin (x)} \sin (x) + \cos (x) \csc (x)$

Schritt 4.2.3.3

Potenziere

\sin (x)

$\sin (x)$ mit

1

$1$ .

- \frac{\cos (x)}{\sin^{1} (x) \sin^{1} (x)} \sin (x) + \cos (x) \csc (x)

$- \frac{\cos (x)}{\sin^{1} (x) \sin^{1} (x)} \sin (x) + \cos (x) \csc (x)$

Schritt 4.2.3.4

Wende die Exponentenregel

a^{m} a^{n} = a^{m + n}

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

- \frac{\cos (x)}{{\sin (x)}^{1 + 1}} \sin (x) + \cos (x) \csc (x)

$- \frac{\cos (x)}{{\sin (x)}^{1 + 1}} \sin (x) + \cos (x) \csc (x)$

Schritt 4.2.3.5

Addiere

1

$1$ und

1

$1$ .

- \frac{\cos (x)}{\sin^{2} (x)} \sin (x) + \cos (x) \csc (x)

$- \frac{\cos (x)}{\sin^{2} (x)} \sin (x) + \cos (x) \csc (x)$

- \frac{\cos (x)}{\sin^{2} (x)} \sin (x) + \cos (x) \csc (x)

$- \frac{\cos (x)}{\sin^{2} (x)} \sin (x) + \cos (x) \csc (x)$

Schritt 4.2.4

Kürze den gemeinsamen Faktor von

\sin (x)

$\sin (x)$ .

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Schritt 4.2.4.1

Bringe das führende Minuszeichen in

- \frac{\cos (x)}{\sin^{2} (x)}

$- \frac{\cos (x)}{\sin^{2} (x)}$ in den Zähler.

\frac{- \cos (x)}{\sin^{2} (x)} \sin (x) + \cos (x) \csc (x)

$\frac{- \cos (x)}{\sin^{2} (x)} \sin (x) + \cos (x) \csc (x)$

Schritt 4.2.4.2

Faktorisiere

\sin (x)

$\sin (x)$ aus

\sin^{2} (x)

$\sin^{2} (x)$ heraus.

\frac{- \cos (x)}{\sin (x) \sin (x)} \sin (x) + \cos (x) \csc (x)

$\frac{- \cos (x)}{\sin (x) \sin (x)} \sin (x) + \cos (x) \csc (x)$

Schritt 4.2.4.3

Kürze den gemeinsamen Faktor.

\frac{- \cos (x)}{\sin (x) \sin (x)} \sin (x) + \cos (x) \csc (x)

$\frac{- \cos (x)}{\sin (x) \sin (x)} \sin (x) + \cos (x) \csc (x)$

Schritt 4.2.4.4

Forme den Ausdruck um.

\frac{- \cos (x)}{\sin (x)} + \cos (x) \csc (x)

$\frac{- \cos (x)}{\sin (x)} + \cos (x) \csc (x)$

\frac{- \cos (x)}{\sin (x)} + \cos (x) \csc (x)

$\frac{- \cos (x)}{\sin (x)} + \cos (x) \csc (x)$

Schritt 4.2.5

Ziehe das Minuszeichen vor den Bruch.

- \frac{\cos (x)}{\sin (x)} + \cos (x) \csc (x)

$- \frac{\cos (x)}{\sin (x)} + \cos (x) \csc (x)$

Schritt 4.2.6

Schreibe

\csc (x)

$\csc (x)$ mithilfe von Sinus und Kosinus um.

- \frac{\cos (x)}{\sin (x)} + \cos (x) \frac{1}{\sin (x)}

$- \frac{\cos (x)}{\sin (x)} + \cos (x) \frac{1}{\sin (x)}$

Schritt 4.2.7

Kombiniere

\cos (x)

$\cos (x)$ und

\frac{1}{\sin (x)}

$\frac{1}{\sin (x)}$ .

- \frac{\cos (x)}{\sin (x)} + \frac{\cos (x)}{\sin (x)}

$- \frac{\cos (x)}{\sin (x)} + \frac{\cos (x)}{\sin (x)}$

- \frac{\cos (x)}{\sin (x)} + \frac{\cos (x)}{\sin (x)}

$- \frac{\cos (x)}{\sin (x)} + \frac{\cos (x)}{\sin (x)}$

Schritt 4.3

Addiere

- \frac{\cos (x)}{\sin (x)}

$- \frac{\cos (x)}{\sin (x)}$ und

\frac{\cos (x)}{\sin (x)}

$\frac{\cos (x)}{\sin (x)}$ .

0

$0$

0

$0$