Cálculo Exemplos

$\csc (x) \tan (x) \cos (x)$

Etapa 1

Diferencie usando a regra do produto, que determina que $\frac{d}{d x} [f (x) g (x)]$ é $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ , em que $f (x) = \csc (x) \tan (x)$ e $g (x) = \cos (x)$ .

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

Etapa 2

A derivada de $\cos (x)$ em relação a $x$ é $- \sin (x)$ .

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

Etapa 3

Diferencie usando a regra do produto, que determina que $\frac{d}{d x} [f (x) g (x)]$ é $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ , em que $f (x) = \csc (x)$ e $g (x) = \tan (x)$ .

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

Etapa 4

A derivada de $\tan (x)$ em relação a $x$ é $\sec^{2} (x)$ .

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

Etapa 5

A derivada de $\csc (x)$ em relação a $x$ é $- \csc (x) \cot (x)$ .

$\csc (x) \tan (x) (- \sin (x)) + \cos (x) (\csc (x) \sec^{2} (x) + \tan (x) (- \csc (x) \cot (x)))$

Etapa 6

Simplifique.

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Etapa 6.1

Aplique a propriedade distributiva.

$\csc (x) \tan (x) (- \sin (x)) + \cos (x) (\csc (x) \sec^{2} (x)) + \cos (x) (\tan (x) (- \csc (x) \cot (x)))$

Etapa 6.2

Remova os parênteses desnecessários.

$\csc (x) \tan (x) (- \sin (x)) + \cos (x) \csc (x) \sec^{2} (x) + \cos (x) \tan (x) (- \csc (x) \cot (x))$

Etapa 6.3

Reordene os termos.

$- \csc (x) \sin (x) \tan (x) + \sec^{2} (x) \cos (x) \csc (x) - \cos (x) \cot (x) \csc (x) \tan (x)$

Etapa 6.4

Simplifique cada termo.

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Etapa 6.4.1

Reescreva em termos de senos e cossenos e, depois, cancele os fatores comuns.

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Etapa 6.4.1.1

Adicione parênteses.

$- (\csc (x) \sin (x)) \tan (x) + \sec^{2} (x) \cos (x) \csc (x) - \cos (x) \cot (x) \csc (x) \tan (x)$

Etapa 6.4.1.2

Reescreva $- \csc (x) \sin (x)$ em termos de senos e cossenos.

$- (\frac{1}{\sin (x)} \sin (x)) \tan (x) + \sec^{2} (x) \cos (x) \csc (x) - \cos (x) \cot (x) \csc (x) \tan (x)$

Etapa 6.4.1.3

Cancele os fatores comuns.

$- 1 \cdot 1 \tan (x) + \sec^{2} (x) \cos (x) \csc (x) - \cos (x) \cot (x) \csc (x) \tan (x)$

Etapa 6.4.2

Multiplique $- 1$ por $1$ .

$- \tan (x) + \sec^{2} (x) \cos (x) \csc (x) - \cos (x) \cot (x) \csc (x) \tan (x)$

Etapa 6.4.3

Reescreva $\tan (x)$ em termos de senos e cossenos.

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

Etapa 6.4.4

Reescreva $\sec (x)$ em termos de senos e cossenos.

$- \frac{\sin (x)}{\cos (x)} + {(\frac{1}{\cos (x)})}^{2} \cos (x) \csc (x) - \cos (x) \cot (x) \csc (x) \tan (x)$

Etapa 6.4.5

Aplique a regra do produto a $\frac{1}{\cos (x)}$ .

$- \frac{\sin (x)}{\cos (x)} + \frac{1^{2}}{\cos^{2} (x)} \cos (x) \csc (x) - \cos (x) \cot (x) \csc (x) \tan (x)$

Etapa 6.4.6

Cancele o fator comum de $\cos (x)$ .

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Etapa 6.4.6.1

Fatore $\cos (x)$ de $\cos^{2} (x)$ .

$- \frac{\sin (x)}{\cos (x)} + \frac{1^{2}}{\cos (x) \cos (x)} \cos (x) \csc (x) - \cos (x) \cot (x) \csc (x) \tan (x)$

Etapa 6.4.6.2

Cancele o fator comum.

$- \frac{\sin (x)}{\cos (x)} + \frac{1^{2}}{\cos (x) \cos (x)} \cos (x) \csc (x) - \cos (x) \cot (x) \csc (x) \tan (x)$

Etapa 6.4.6.3

Reescreva a expressão.

$- \frac{\sin (x)}{\cos (x)} + \frac{1^{2}}{\cos (x)} \csc (x) - \cos (x) \cot (x) \csc (x) \tan (x)$

Etapa 6.4.7

Um elevado a qualquer potência é um.

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

Etapa 6.4.8

Reescreva $\csc (x)$ em termos de senos e cossenos.

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

Etapa 6.4.9

Multiplique $\frac{1}{\cos (x)}$ por $\frac{1}{\sin (x)}$ .

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

Etapa 6.4.10

Reescreva $\cot (x)$ em termos de senos e cossenos.

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

Etapa 6.4.11

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

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Etapa 6.4.11.1

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

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

Etapa 6.4.11.2

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 6.4.11.3

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 6.4.11.4

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

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

Etapa 6.4.11.5

Some $1$ e $1$ .

$- \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x) \sin (x)} - \frac{\cos^{2} (x)}{\sin (x)} \csc (x) \tan (x)$

Etapa 6.4.12

Reescreva $\csc (x)$ em termos de senos e cossenos.

$- \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x) \sin (x)} - \frac{\cos^{2} (x)}{\sin (x)} \cdot \frac{1}{\sin (x)} \tan (x)$

Etapa 6.4.13

Multiplique $- \frac{\cos^{2} (x)}{\sin (x)} \cdot \frac{1}{\sin (x)}$ .

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Etapa 6.4.13.1

Multiplique $\frac{1}{\sin (x)}$ por $\frac{\cos^{2} (x)}{\sin (x)}$ .

$- \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x) \sin (x)} - \frac{\cos^{2} (x)}{\sin (x) \sin (x)} \tan (x)$

Etapa 6.4.13.2

Eleve $\sin (x)$ à potência de $1$ .

$- \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x) \sin (x)} - \frac{\cos^{2} (x)}{\sin^{1} (x) \sin (x)} \tan (x)$

Etapa 6.4.13.3

Eleve $\sin (x)$ à potência de $1$ .

$- \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x) \sin (x)} - \frac{\cos^{2} (x)}{\sin^{1} (x) \sin^{1} (x)} \tan (x)$

Etapa 6.4.13.4

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

$- \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x) \sin (x)} - \frac{\cos^{2} (x)}{{\sin (x)}^{1 + 1}} \tan (x)$

Etapa 6.4.13.5

Some $1$ e $1$ .

$- \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x) \sin (x)} - \frac{\cos^{2} (x)}{\sin^{2} (x)} \tan (x)$

Etapa 6.4.14

Reescreva $\tan (x)$ em termos de senos e cossenos.

$- \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x) \sin (x)} - \frac{\cos^{2} (x)}{\sin^{2} (x)} \cdot \frac{\sin (x)}{\cos (x)}$

Etapa 6.4.15

Cancele o fator comum de $\cos (x)$ .

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Etapa 6.4.15.1

Mova o negativo de maior ordem em $- \frac{\cos^{2} (x)}{\sin^{2} (x)}$ para o numerador.

$- \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x) \sin (x)} + \frac{- \cos^{2} (x)}{\sin^{2} (x)} \cdot \frac{\sin (x)}{\cos (x)}$

Etapa 6.4.15.2

Fatore $\cos (x)$ de $- \cos^{2} (x)$ .

$- \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x) \sin (x)} + \frac{\cos (x) (- \cos (x))}{\sin^{2} (x)} \cdot \frac{\sin (x)}{\cos (x)}$

Etapa 6.4.15.3

Cancele o fator comum.

$- \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x) \sin (x)} + \frac{\cos (x) (- \cos (x))}{\sin^{2} (x)} \cdot \frac{\sin (x)}{\cos (x)}$

Etapa 6.4.15.4

Reescreva a expressão.

$- \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x) \sin (x)} + \frac{- \cos (x)}{\sin^{2} (x)} \sin (x)$

Etapa 6.4.16

Cancele o fator comum de $\sin (x)$ .

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Etapa 6.4.16.1

Fatore $\sin (x)$ de $\sin^{2} (x)$ .

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

Etapa 6.4.16.2

Cancele o fator comum.

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

Etapa 6.4.16.3

Reescreva a expressão.

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

Etapa 6.4.17

Mova o número negativo para a frente da fração.

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

Etapa 6.5

Simplifique cada termo.

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Etapa 6.5.1

Converta de $\frac{\sin (x)}{\cos (x)}$ em $\tan (x)$ .

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

Etapa 6.5.2

Separe as frações.

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

Etapa 6.5.3

Converta de $\frac{1}{\cos (x)}$ em $\sec (x)$ .

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

Etapa 6.5.4

Converta de $\frac{1}{\sin (x)}$ em $\csc (x)$ .

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

Etapa 6.5.5

Converta de $\frac{\cos (x)}{\sin (x)}$ em $\cot (x)$ .

$- \tan (x) + \csc (x) \sec (x) - \cot (x)$