Cálculo Exemplos

$y = (x^{2} + \sin (x)) \sec (x)$

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 $f (x) = x^{2} + \sin (x)$ e $g (x) = \sec (x)$ .

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

Etapa 2

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

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

Etapa 3

Diferencie.

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Etapa 3.1

De acordo com a regra da soma, a derivada de $x^{2} + \sin (x)$ com relação a $x$ é $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [\sin (x)]$ .

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

Etapa 3.2

Diferencie usando a regra da multiplicação de potências, que determina que $\frac{d}{d x} [x^{n}]$ é $n x^{n - 1}$ , em que $n = 2$ .

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

Etapa 4

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

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

Etapa 5

Simplifique.

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Etapa 5.1

Aplique a propriedade distributiva.

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

Etapa 5.2

Aplique a propriedade distributiva.

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

Etapa 5.3

Aplique a propriedade distributiva.

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

Etapa 5.4

Reordene os termos.

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

Etapa 5.5

Simplifique cada termo.

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Etapa 5.5.1

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

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

Etapa 5.5.2

Combine $x^{2}$ e $\frac{1}{\cos (x)}$ .

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

Etapa 5.5.3

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

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

Etapa 5.5.4

Combine.

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

Etapa 5.5.5

Simplifique o denominador.

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Etapa 5.5.5.1

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 5.5.5.2

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 5.5.5.3

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

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

Etapa 5.5.5.4

Some $1$ e $1$ .

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

Etapa 5.5.6

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

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

Etapa 5.5.7

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

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

Etapa 5.5.8

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

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

Etapa 5.5.9

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

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Etapa 5.5.9.1

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

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

Etapa 5.5.9.2

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 5.5.9.3

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 5.5.9.4

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

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

Etapa 5.5.9.5

Some $1$ e $1$ .

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

Etapa 5.5.9.6

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 5.5.9.7

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 5.5.9.8

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

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

Etapa 5.5.9.9

Some $1$ e $1$ .

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

Etapa 5.5.10

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

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

Etapa 5.5.11

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

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Etapa 5.5.11.1

Combine $2$ e $\frac{1}{\cos (x)}$ .

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

Etapa 5.5.11.2

Combine $x$ e $\frac{2}{\cos (x)}$ .

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

Etapa 5.5.12

Mova $2$ para a esquerda de $x$ .

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

Etapa 5.5.13

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

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Etapa 5.5.13.1

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

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

Etapa 5.5.13.2

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

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

Etapa 5.5.13.3

Cancele os fatores comuns.

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

Etapa 5.6

Simplifique cada termo.

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Etapa 5.6.1

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

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

Etapa 5.6.2

Separe as frações.

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

Etapa 5.6.3

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

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

Etapa 5.6.4

Separe as frações.

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

Etapa 5.6.5

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

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

Etapa 5.6.6

Divida $x^{2}$ por $1$ .

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

Etapa 5.6.7

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

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

Etapa 5.7

Mova $1$ .

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

Etapa 5.8

Aplique a identidade trigonométrica fundamental.

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