Cálculo Exemplos

$f (x) = (\sin (x) + \cos (x)) \sec (x)$

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 $f (x) = \sin (x) + \cos (x)$ e $g (x) = \sec (x)$ .

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

Etapa 2

A derivada de $\sec (x)$ em relação a $x$ é $\sec (x) \tan (x)$ .

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

Etapa 3

De acordo com a regra da soma, a derivada de $\sin (x) + \cos (x)$ com relação a $x$ é $\frac{d}{d x} [\sin (x)] + \frac{d}{d x} [\cos (x)]$ .

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

Etapa 4

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

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

Etapa 5

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

$(\sin (x) + \cos (x)) \sec (x) \tan (x) + \sec (x) (\cos (x) - \sin (x))$

Etapa 6

Simplifique.

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Etapa 6.1

Aplique a propriedade distributiva.

$(\sin (x) \sec (x) + \cos (x) \sec (x)) \tan (x) + \sec (x) (\cos (x) - \sin (x))$

Etapa 6.2

Aplique a propriedade distributiva.

$\sin (x) \sec (x) \tan (x) + \cos (x) \sec (x) \tan (x) + \sec (x) (\cos (x) - \sin (x))$

Etapa 6.3

Aplique a propriedade distributiva.

$\sin (x) \sec (x) \tan (x) + \cos (x) \sec (x) \tan (x) + \sec (x) \cos (x) + \sec (x) (- \sin (x))$

Etapa 6.4

Reordene os termos.

$\sec (x) \sin (x) \tan (x) + \cos (x) \sec (x) \tan (x) + \cos (x) \sec (x) - \sec (x) \sin (x)$

Etapa 6.5

Simplifique cada termo.

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Etapa 6.5.1

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

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

Etapa 6.5.2

Combine $\frac{1}{\cos (x)}$ e $\sin (x)$ .

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

Etapa 6.5.3

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

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

Etapa 6.5.4

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

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Etapa 6.5.4.1

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

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

Etapa 6.5.4.2

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 6.5.4.3

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 6.5.4.4

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

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

Etapa 6.5.4.5

Some $1$ e $1$ .

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

Etapa 6.5.4.6

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 6.5.4.7

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 6.5.4.8

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

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

Etapa 6.5.4.9

Some $1$ e $1$ .

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

Etapa 6.5.5

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

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Etapa 6.5.5.1

Reordene $\cos (x)$ e $\sec (x)$ .

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

Etapa 6.5.5.2

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

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

Etapa 6.5.5.3

Cancele os fatores comuns.

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

Etapa 6.5.6

Multiplique $\tan (x)$ por $1$ .

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

Etapa 6.5.7

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

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

Etapa 6.5.8

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

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Etapa 6.5.8.1

Reordene $\cos (x)$ e $\sec (x)$ .

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

Etapa 6.5.8.2

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

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

Etapa 6.5.8.3

Cancele os fatores comuns.

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

Etapa 6.5.9

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

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

Etapa 6.5.10

Combine $\sin (x)$ e $\frac{1}{\cos (x)}$ .

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

Etapa 6.6

Combine os termos opostos em $\frac{\sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin (x)}{\cos (x)} + 1 - \frac{\sin (x)}{\cos (x)}$ .

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Etapa 6.6.1

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

$\frac{\sin^{2} (x)}{\cos^{2} (x)} + 0 + 1$

Etapa 6.6.2

Some $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ e $0$ .

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

Etapa 6.7

Converta de $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ em $\tan^{2} (x)$ .

$\tan^{2} (x) + 1$

Etapa 6.8

Aplique a identidade trigonométrica fundamental.

$\sec^{2} (x)$