Cálculo Exemplos

$f (x) = \ln (x^{4} (\cos (x) + \sin (x)))$

Etapa 1

Diferencie usando a regra da cadeia, que determina que $\frac{d}{d x} [f (g (x))]$ é $f' (g (x)) g' (x)$ , em que $f (x) = \ln (x)$ e $g (x) = x^{4} (\cos (x) + \sin (x))$ .

Toque para ver mais passagens...

Etapa 1.1

Para aplicar a regra da cadeia, defina $u$ como $x^{4} (\cos (x) + \sin (x))$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x^{4} (\cos (x) + \sin (x))]$

Etapa 1.2

A derivada de $\ln (u)$ em relação a $u$ é $\frac{1}{u}$ .

$\frac{1}{u} \frac{d}{d x} [x^{4} (\cos (x) + \sin (x))]$

Etapa 1.3

Substitua todas as ocorrências de $u$ por $x^{4} (\cos (x) + \sin (x))$ .

$\frac{1}{x^{4} (\cos (x) + \sin (x))} \frac{d}{d x} [x^{4} (\cos (x) + \sin (x))]$

Etapa 2

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 $f (x) = x^{4}$ e $g (x) = \cos (x) + \sin (x)$ .

$\frac{1}{x^{4} (\cos (x) + \sin (x))} (x^{4} \frac{d}{d x} [\cos (x) + \sin (x)] + (\cos (x) + \sin (x)) \frac{d}{d x} [x^{4}])$

Etapa 3

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

$\frac{1}{x^{4} (\cos (x) + \sin (x))} (x^{4} (\frac{d}{d x} [\cos (x)] + \frac{d}{d x} [\sin (x)]) + (\cos (x) + \sin (x)) \frac{d}{d x} [x^{4}])$

Etapa 4

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

$\frac{1}{x^{4} (\cos (x) + \sin (x))} (x^{4} (- \sin (x) + \frac{d}{d x} [\sin (x)]) + (\cos (x) + \sin (x)) \frac{d}{d x} [x^{4}])$

Etapa 5

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

$\frac{1}{x^{4} (\cos (x) + \sin (x))} (x^{4} (- \sin (x) + \cos (x)) + (\cos (x) + \sin (x)) \frac{d}{d x} [x^{4}])$

Etapa 6

Diferencie usando a regra da multiplicação de potências.

Toque para ver mais passagens...

Etapa 6.1

Diferencie usando a regra da multiplicação de potências, que determina que $\frac{d}{d x} [x^{n}]$ é $n x^{n - 1}$ , em que $n = 4$ .

$\frac{1}{x^{4} (\cos (x) + \sin (x))} (x^{4} (- \sin (x) + \cos (x)) + (\cos (x) + \sin (x)) (4 x^{3}))$

Etapa 6.2

Mova $4$ para a esquerda de $\cos (x) + \sin (x)$ .

$\frac{1}{x^{4} (\cos (x) + \sin (x))} (x^{4} (- \sin (x) + \cos (x)) + 4 \cdot (\cos (x) + \sin (x)) x^{3})$

Etapa 7

Simplifique.

Toque para ver mais passagens...

Etapa 7.1

Aplique a propriedade distributiva.

$\frac{1}{x^{4} \cos (x) + x^{4} \sin (x)} (x^{4} (- \sin (x) + \cos (x)) + 4 (\cos (x) + \sin (x)) x^{3})$

Etapa 7.2

Aplique a propriedade distributiva.

$\frac{1}{x^{4} \cos (x) + x^{4} \sin (x)} (x^{4} (- \sin (x)) + x^{4} \cos (x) + 4 (\cos (x) + \sin (x)) x^{3})$

Etapa 7.3

Aplique a propriedade distributiva.

$\frac{1}{x^{4} \cos (x) + x^{4} \sin (x)} (x^{4} (- \sin (x)) + x^{4} \cos (x) + (4 \cos (x) + 4 \sin (x)) x^{3})$

Etapa 7.4

Aplique a propriedade distributiva.

$\frac{1}{x^{4} \cos (x) + x^{4} \sin (x)} (x^{4} (- \sin (x)) + x^{4} \cos (x) + 4 \cos (x) x^{3} + 4 \sin (x) x^{3})$

Etapa 7.5

Reordene os fatores de $\frac{1}{x^{4} \cos (x) + x^{4} \sin (x)} (x^{4} (- \sin (x)) + x^{4} \cos (x) + 4 \cos (x) x^{3} + 4 \sin (x) x^{3})$ .

$(x^{4} (- \sin (x)) + x^{4} \cos (x) + 4 \cos (x) x^{3} + 4 \sin (x) x^{3}) \frac{1}{x^{4} \cos (x) + x^{4} \sin (x)}$

Etapa 7.6

Fatore $x^{4}$ de $x^{4} \cos (x) + x^{4} \sin (x)$ .

Toque para ver mais passagens...

Etapa 7.6.1

Fatore $x^{4}$ de $x^{4} \cos (x)$ .

$(x^{4} (- \sin (x)) + x^{4} \cos (x) + 4 \cos (x) x^{3} + 4 \sin (x) x^{3}) \frac{1}{x^{4} (\cos (x)) + x^{4} \sin (x)}$

Etapa 7.6.2

Fatore $x^{4}$ de $x^{4} \sin (x)$ .

$(x^{4} (- \sin (x)) + x^{4} \cos (x) + 4 \cos (x) x^{3} + 4 \sin (x) x^{3}) \frac{1}{x^{4} (\cos (x)) + x^{4} (\sin (x))}$

Etapa 7.6.3

Fatore $x^{4}$ de $x^{4} (\cos (x)) + x^{4} (\sin (x))$ .

$(x^{4} (- \sin (x)) + x^{4} \cos (x) + 4 \cos (x) x^{3} + 4 \sin (x) x^{3}) \frac{1}{x^{4} (\cos (x) + \sin (x))}$

Etapa 7.7

Reescreva usando a propriedade comutativa da multiplicação.

$(- x^{4} \sin (x) + x^{4} \cos (x) + 4 \cos (x) x^{3} + 4 \sin (x) x^{3}) \frac{1}{x^{4} (\cos (x) + \sin (x))}$

Etapa 7.8

Aplique a propriedade distributiva.

$- x^{4} \sin (x) \frac{1}{x^{4} (\cos (x) + \sin (x))} + x^{4} \cos (x) \frac{1}{x^{4} (\cos (x) + \sin (x))} + 4 \cos (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))} + 4 \sin (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))}$

Etapa 7.9

Simplifique.

Toque para ver mais passagens...

Etapa 7.9.1

Cancele o fator comum de $x^{4}$ .

Toque para ver mais passagens...

Etapa 7.9.1.1

Fatore $x^{4}$ de $- x^{4} \sin (x)$ .

$x^{4} (- \sin (x)) \frac{1}{x^{4} (\cos (x) + \sin (x))} + x^{4} \cos (x) \frac{1}{x^{4} (\cos (x) + \sin (x))} + 4 \cos (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))} + 4 \sin (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))}$

Etapa 7.9.1.2

Cancele o fator comum.

Etapa 7.9.1.3

Reescreva a expressão.

$- \sin (x) \frac{1}{\cos (x) + \sin (x)} + x^{4} \cos (x) \frac{1}{x^{4} (\cos (x) + \sin (x))} + 4 \cos (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))} + 4 \sin (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))}$

Etapa 7.9.2

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

$- \frac{\sin (x)}{\cos (x) + \sin (x)} + x^{4} \cos (x) \frac{1}{x^{4} (\cos (x) + \sin (x))} + 4 \cos (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))} + 4 \sin (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))}$

Etapa 7.9.3

Cancele o fator comum de $x^{4}$ .

Toque para ver mais passagens...

Etapa 7.9.3.1

Fatore $x^{4}$ de $x^{4} \cos (x)$ .

$- \frac{\sin (x)}{\cos (x) + \sin (x)} + x^{4} (\cos (x)) \frac{1}{x^{4} (\cos (x) + \sin (x))} + 4 \cos (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))} + 4 \sin (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))}$

Etapa 7.9.3.2

Cancele o fator comum.

Etapa 7.9.3.3

Reescreva a expressão.

$- \frac{\sin (x)}{\cos (x) + \sin (x)} + \cos (x) \frac{1}{\cos (x) + \sin (x)} + 4 \cos (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))} + 4 \sin (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))}$

Etapa 7.9.4

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

$- \frac{\sin (x)}{\cos (x) + \sin (x)} + \frac{\cos (x)}{\cos (x) + \sin (x)} + 4 \cos (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))} + 4 \sin (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))}$

Etapa 7.9.5

Cancele o fator comum de $x^{3}$ .

Toque para ver mais passagens...

Etapa 7.9.5.1

Fatore $x^{3}$ de $4 \cos (x) x^{3}$ .

$- \frac{\sin (x)}{\cos (x) + \sin (x)} + \frac{\cos (x)}{\cos (x) + \sin (x)} + x^{3} (4 \cos (x)) \frac{1}{x^{4} (\cos (x) + \sin (x))} + 4 \sin (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))}$

Etapa 7.9.5.2

Fatore $x^{3}$ de $x^{4} (\cos (x) + \sin (x))$ .

$- \frac{\sin (x)}{\cos (x) + \sin (x)} + \frac{\cos (x)}{\cos (x) + \sin (x)} + x^{3} (4 \cos (x)) \frac{1}{x^{3} (x (\cos (x) + \sin (x)))} + 4 \sin (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))}$

Etapa 7.9.5.3

Cancele o fator comum.

$- \frac{\sin (x)}{\cos (x) + \sin (x)} + \frac{\cos (x)}{\cos (x) + \sin (x)} + x^{3} (4 \cos (x)) \frac{1}{x^{3} (x (\cos (x) + \sin (x)))} + 4 \sin (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))}$

Etapa 7.9.5.4

Reescreva a expressão.

$- \frac{\sin (x)}{\cos (x) + \sin (x)} + \frac{\cos (x)}{\cos (x) + \sin (x)} + 4 \cos (x) \frac{1}{x (\cos (x) + \sin (x))} + 4 \sin (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))}$

Etapa 7.9.6

Combine $\frac{1}{x (\cos (x) + \sin (x))}$ e $4$ .

$- \frac{\sin (x)}{\cos (x) + \sin (x)} + \frac{\cos (x)}{\cos (x) + \sin (x)} + \frac{4}{x (\cos (x) + \sin (x))} \cos (x) + 4 \sin (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))}$

Etapa 7.9.7

Combine $\frac{4}{x (\cos (x) + \sin (x))}$ e $\cos (x)$ .

$- \frac{\sin (x)}{\cos (x) + \sin (x)} + \frac{\cos (x)}{\cos (x) + \sin (x)} + \frac{4 \cos (x)}{x (\cos (x) + \sin (x))} + 4 \sin (x) x^{3} \frac{1}{x^{4} (\cos (x) + \sin (x))}$

Etapa 7.9.8

Cancele o fator comum de $x^{3}$ .

Toque para ver mais passagens...

Etapa 7.9.8.1

Fatore $x^{3}$ de $4 \sin (x) x^{3}$ .

$- \frac{\sin (x)}{\cos (x) + \sin (x)} + \frac{\cos (x)}{\cos (x) + \sin (x)} + \frac{4 \cos (x)}{x (\cos (x) + \sin (x))} + x^{3} (4 \sin (x)) \frac{1}{x^{4} (\cos (x) + \sin (x))}$

Etapa 7.9.8.2

Fatore $x^{3}$ de $x^{4} (\cos (x) + \sin (x))$ .

$- \frac{\sin (x)}{\cos (x) + \sin (x)} + \frac{\cos (x)}{\cos (x) + \sin (x)} + \frac{4 \cos (x)}{x (\cos (x) + \sin (x))} + x^{3} (4 \sin (x)) \frac{1}{x^{3} (x (\cos (x) + \sin (x)))}$

Etapa 7.9.8.3

Cancele o fator comum.

$- \frac{\sin (x)}{\cos (x) + \sin (x)} + \frac{\cos (x)}{\cos (x) + \sin (x)} + \frac{4 \cos (x)}{x (\cos (x) + \sin (x))} + x^{3} (4 \sin (x)) \frac{1}{x^{3} (x (\cos (x) + \sin (x)))}$

Etapa 7.9.8.4

Reescreva a expressão.

$- \frac{\sin (x)}{\cos (x) + \sin (x)} + \frac{\cos (x)}{\cos (x) + \sin (x)} + \frac{4 \cos (x)}{x (\cos (x) + \sin (x))} + 4 \sin (x) \frac{1}{x (\cos (x) + \sin (x))}$

Etapa 7.9.9

Combine $\frac{1}{x (\cos (x) + \sin (x))}$ e $4$ .

$- \frac{\sin (x)}{\cos (x) + \sin (x)} + \frac{\cos (x)}{\cos (x) + \sin (x)} + \frac{4 \cos (x)}{x (\cos (x) + \sin (x))} + \frac{4}{x (\cos (x) + \sin (x))} \sin (x)$

Etapa 7.9.10

Combine $\frac{4}{x (\cos (x) + \sin (x))}$ e $\sin (x)$ .

$- \frac{\sin (x)}{\cos (x) + \sin (x)} + \frac{\cos (x)}{\cos (x) + \sin (x)} + \frac{4 \cos (x)}{x (\cos (x) + \sin (x))} + \frac{4 \sin (x)}{x (\cos (x) + \sin (x))}$

Etapa 7.10

Combine os numeradores em relação ao denominador comum.

$\frac{- \sin (x) + \cos (x)}{\cos (x) + \sin (x)} + \frac{4 \cos (x)}{x (\cos (x) + \sin (x))} + \frac{4 \sin (x)}{x (\cos (x) + \sin (x))}$

Etapa 7.11

Para escrever $\frac{- \sin (x) + \cos (x)}{\cos (x) + \sin (x)}$ como fração com um denominador comum, multiplique por $\frac{x}{x}$ .

$\frac{- \sin (x) + \cos (x)}{\cos (x) + \sin (x)} \cdot \frac{x}{x} + \frac{4 \cos (x)}{x (\cos (x) + \sin (x))} + \frac{4 \sin (x)}{x (\cos (x) + \sin (x))}$

Etapa 7.12

Escreva cada expressão com um denominador comum de $(\cos (x) + \sin (x)) x$ , multiplicando cada um por um fator apropriado de $1$ .

Toque para ver mais passagens...

Etapa 7.12.1

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

$\frac{(- \sin (x) + \cos (x)) x}{(\cos (x) + \sin (x)) x} + \frac{4 \cos (x)}{x (\cos (x) + \sin (x))} + \frac{4 \sin (x)}{x (\cos (x) + \sin (x))}$

Etapa 7.12.2

Reordene os fatores de $(\cos (x) + \sin (x)) x$ .

$\frac{(- \sin (x) + \cos (x)) x}{x (\cos (x) + \sin (x))} + \frac{4 \cos (x)}{x (\cos (x) + \sin (x))} + \frac{4 \sin (x)}{x (\cos (x) + \sin (x))}$

Etapa 7.13

Combine os numeradores em relação ao denominador comum.

$\frac{(- \sin (x) + \cos (x)) x + 4 \cos (x)}{x (\cos (x) + \sin (x))} + \frac{4 \sin (x)}{x (\cos (x) + \sin (x))}$

Etapa 7.14

Aplique a propriedade distributiva.

$\frac{- \sin (x) x + \cos (x) x + 4 \cos (x)}{x (\cos (x) + \sin (x))} + \frac{4 \sin (x)}{x (\cos (x) + \sin (x))}$

Etapa 7.15

Combine os numeradores em relação ao denominador comum.

$\frac{- \sin (x) x + \cos (x) x + 4 \cos (x) + 4 \sin (x)}{x (\cos (x) + \sin (x))}$

Etapa 7.16

Fatore $- 1$ de $- \sin (x) x$ .

$\frac{- (\sin (x) x) + \cos (x) x + 4 \cos (x) + 4 \sin (x)}{x (\cos (x) + \sin (x))}$

Etapa 7.17

Fatore $- 1$ de $\cos (x) x$ .

$\frac{- (\sin (x) x) - (- \cos (x) x) + 4 \cos (x) + 4 \sin (x)}{x (\cos (x) + \sin (x))}$

Etapa 7.18

Fatore $- 1$ de $- (\sin (x) x) - (- \cos (x) x)$ .

$\frac{- (\sin (x) x - \cos (x) x) + 4 \cos (x) + 4 \sin (x)}{x (\cos (x) + \sin (x))}$

Etapa 7.19

Fatore $- 1$ de $4 \cos (x)$ .

$\frac{- (\sin (x) x - \cos (x) x) - (- 4 \cos (x)) + 4 \sin (x)}{x (\cos (x) + \sin (x))}$

Etapa 7.20

Fatore $- 1$ de $- (\sin (x) x - \cos (x) x) - (- 4 \cos (x))$ .

$\frac{- (\sin (x) x - \cos (x) x - 4 \cos (x)) + 4 \sin (x)}{x (\cos (x) + \sin (x))}$

Etapa 7.21

Fatore $- 1$ de $4 \sin (x)$ .

$\frac{- (\sin (x) x - \cos (x) x - 4 \cos (x)) - (- 4 \sin (x))}{x (\cos (x) + \sin (x))}$

Etapa 7.22

Fatore $- 1$ de $- (\sin (x) x - \cos (x) x - 4 \cos (x)) - (- 4 \sin (x))$ .

$\frac{- (\sin (x) x - \cos (x) x - 4 \cos (x) - 4 \sin (x))}{x (\cos (x) + \sin (x))}$

Etapa 7.23

Reescreva $- (\sin (x) x - \cos (x) x - 4 \cos (x) - 4 \sin (x))$ como $- 1 (\sin (x) x - \cos (x) x - 4 \cos (x) - 4 \sin (x))$ .

$\frac{- 1 (\sin (x) x - \cos (x) x - 4 \cos (x) - 4 \sin (x))}{x (\cos (x) + \sin (x))}$

Etapa 7.24

Mova o número negativo para a frente da fração.

$- \frac{\sin (x) x - \cos (x) x - 4 \cos (x) - 4 \sin (x)}{x (\cos (x) + \sin (x))}$

Etapa 7.25

Reordene os fatores em $- \frac{\sin (x) x - \cos (x) x - 4 \cos (x) - 4 \sin (x)}{x (\cos (x) + \sin (x))}$ .

$- \frac{x \sin (x) - x \cos (x) - 4 \cos (x) - 4 \sin (x)}{x (\cos (x) + \sin (x))}$