Cálculo Exemplos

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

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

Etapa 1

Diferencie usando a regra do produto, que determina que

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

$\frac{d}{d x} [f (x) g (x)]$ é

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

$f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ , em que

f (x) = sin (x)

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

g (x) = csc (x)

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

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

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

Etapa 2

A derivada de

csc (x)

$\csc (x)$ em relação a

x

$x$ é

- csc (x) cot (x)

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

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

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

Etapa 3

A derivada de

sin (x)

$\sin (x)$ em relação a

x

$x$ é

cos (x)

$\cos (x)$ .

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

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

Etapa 4

Simplifique.

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Etapa 4.1

Reordene os termos.

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

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

Etapa 4.2

Simplifique cada termo.

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Etapa 4.2.1

Reescreva

cot (x)

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

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

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

Etapa 4.2.2

Reescreva

csc (x)

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

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

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

Etapa 4.2.3

Multiplique

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

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

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Etapa 4.2.3.1

Multiplique

\frac{1}{sin (x)}

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

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

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

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

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

Etapa 4.2.3.2

Eleve

sin (x)

$\sin (x)$ à potência de

1

$1$ .

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

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

Etapa 4.2.3.3

Eleve

$\sin (x)$ à potência de

$1$ .

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

Etapa 4.2.3.4

Use a regra da multiplicação de potências

$a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

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

Etapa 4.2.3.5

Some

$1$ e

$1$ .

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

Etapa 4.2.4

Cancele o fator comum de

$\sin (x)$ .

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Etapa 4.2.4.1

Mova o negativo de maior ordem em

$- \frac{\cos (x)}{\sin^{2} (x)}$ para o numerador.

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

Etapa 4.2.4.2

Fatore

$\sin (x)$ de

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

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

Etapa 4.2.4.3

Cancele o fator comum.

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

Etapa 4.2.4.4

Reescreva a expressão.

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

Etapa 4.2.5

Mova o número negativo para a frente da fração.

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

Etapa 4.2.6

Reescreva

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

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

Etapa 4.2.7

Combine

$\cos (x)$ e

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

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

Etapa 4.3

Some

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

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

$0$