Cálculo Exemplos

$y = 5 \sec (x) \csc (x)$

Etapa 1

Encontre a primeira derivada.

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Etapa 1.1

Como $5$ é constante em relação a $x$ , a derivada de $5 \sec (x) \csc (x)$ em relação a $x$ é $5 \frac{d}{d x} [\sec (x) \csc (x)]$ .

$5 \frac{d}{d x} [\sec (x) \csc (x)]$

Etapa 1.2

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 $f (x) = \sec (x)$ e $g (x) = \csc (x)$ .

$5 (\sec (x) \frac{d}{d x} [\csc (x)] + \csc (x) \frac{d}{d x} [\sec (x)])$

Etapa 1.3

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

$5 (\sec (x) (- \csc (x) \cot (x)) + \csc (x) \frac{d}{d x} [\sec (x)])$

Etapa 1.4

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

$5 (\sec (x) (- \csc (x) \cot (x)) + \csc (x) (\sec (x) \tan (x)))$

Etapa 1.5

Simplifique.

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Etapa 1.5.1

Aplique a propriedade distributiva.

$5 (\sec (x) (- \csc (x) \cot (x))) + 5 (\csc (x) \sec (x) \tan (x))$

Etapa 1.5.2

Multiplique $- 1$ por $5$ .

$- 5 \sec (x) \csc (x) \cot (x) + 5 \csc (x) \sec (x) \tan (x)$

Etapa 1.5.3

Reordene os termos.

$f' (x) = - 5 \cot (x) \csc (x) \sec (x) + 5 \csc (x) \sec (x) \tan (x)$

Etapa 2

Encontre a segunda derivada.

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Etapa 2.1

De acordo com a regra da soma, a derivada de $- 5 \cot (x) \csc (x) \sec (x) + 5 \csc (x) \sec (x) \tan (x)$ com relação a $x$ é $\frac{d}{d x} [- 5 \cot (x) \csc (x) \sec (x)] + \frac{d}{d x} [5 \csc (x) \sec (x) \tan (x)]$ .

$\frac{d}{d x} [- 5 \cot (x) \csc (x) \sec (x)] + \frac{d}{d x} [5 \csc (x) \sec (x) \tan (x)]$

Etapa 2.2

Avalie $\frac{d}{d x} [- 5 \cot (x) \csc (x) \sec (x)]$ .

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Etapa 2.2.1

Como $- 5$ é constante em relação a $x$ , a derivada de $- 5 \cot (x) \csc (x) \sec (x)$ em relação a $x$ é $- 5 \frac{d}{d x} [\cot (x) \csc (x) \sec (x)]$ .

$- 5 \frac{d}{d x} [\cot (x) \csc (x) \sec (x)] + \frac{d}{d x} [5 \csc (x) \sec (x) \tan (x)]$

Etapa 2.2.2

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 $f (x) = \cot (x) \csc (x)$ e $g (x) = \sec (x)$ .

$- 5 (\cot (x) \csc (x) \frac{d}{d x} [\sec (x)] + \sec (x) \frac{d}{d x} [\cot (x) \csc (x)]) + \frac{d}{d x} [5 \csc (x) \sec (x) \tan (x)]$

Etapa 2.2.3

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

$- 5 (\cot (x) \csc (x) (\sec (x) \tan (x)) + \sec (x) \frac{d}{d x} [\cot (x) \csc (x)]) + \frac{d}{d x} [5 \csc (x) \sec (x) \tan (x)]$

Etapa 2.2.4

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 $f (x) = \cot (x)$ e $g (x) = \csc (x)$ .

$- 5 (\cot (x) \csc (x) (\sec (x) \tan (x)) + \sec (x) (\cot (x) \frac{d}{d x} [\csc (x)] + \csc (x) \frac{d}{d x} [\cot (x)])) + \frac{d}{d x} [5 \csc (x) \sec (x) \tan (x)]$

Etapa 2.2.5

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

$- 5 (\cot (x) \csc (x) (\sec (x) \tan (x)) + \sec (x) (\cot (x) (- \csc (x) \cot (x)) + \csc (x) \frac{d}{d x} [\cot (x)])) + \frac{d}{d x} [5 \csc (x) \sec (x) \tan (x)]$

Etapa 2.2.6

A derivada de $\cot (x)$ em relação a $x$ é $- \csc^{2} (x)$ .

$- 5 (\cot (x) \csc (x) (\sec (x) \tan (x)) + \sec (x) (\cot (x) (- \csc (x) \cot (x)) + \csc (x) (- \csc^{2} (x)))) + \frac{d}{d x} [5 \csc (x) \sec (x) \tan (x)]$

Etapa 2.2.7

Eleve $\cot (x)$ à potência de $1$ .

$- 5 (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) (\cot^{1} (x) \cot (x)) + \csc (x) (- \csc^{2} (x)))) + \frac{d}{d x} [5 \csc (x) \sec (x) \tan (x)]$

Etapa 2.2.8

Eleve $\cot (x)$ à potência de $1$ .

$- 5 (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) (\cot^{1} (x) \cot^{1} (x)) + \csc (x) (- \csc^{2} (x)))) + \frac{d}{d x} [5 \csc (x) \sec (x) \tan (x)]$

Etapa 2.2.9

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

$- 5 (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) {\cot (x)}^{1 + 1} + \csc (x) (- \csc^{2} (x)))) + \frac{d}{d x} [5 \csc (x) \sec (x) \tan (x)]$

Etapa 2.2.10

Some $1$ e $1$ .

$- 5 (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) + \csc (x) (- \csc^{2} (x)))) + \frac{d}{d x} [5 \csc (x) \sec (x) \tan (x)]$

Etapa 2.2.11

Eleve $\csc (x)$ à potência de $1$ .

$- 5 (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) - (\csc^{1} (x) \csc^{2} (x)))) + \frac{d}{d x} [5 \csc (x) \sec (x) \tan (x)]$

Etapa 2.2.12

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

$- 5 (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) - {\csc (x)}^{1 + 2})) + \frac{d}{d x} [5 \csc (x) \sec (x) \tan (x)]$

Etapa 2.2.13

Some $1$ e $2$ .

$- 5 (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) - \csc^{3} (x))) + \frac{d}{d x} [5 \csc (x) \sec (x) \tan (x)]$

Etapa 2.3

Avalie $\frac{d}{d x} [5 \csc (x) \sec (x) \tan (x)]$ .

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Etapa 2.3.1

Como $5$ é constante em relação a $x$ , a derivada de $5 \csc (x) \sec (x) \tan (x)$ em relação a $x$ é $5 \frac{d}{d x} [\csc (x) \sec (x) \tan (x)]$ .

$- 5 (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) - \csc^{3} (x))) + 5 \frac{d}{d x} [\csc (x) \sec (x) \tan (x)]$

Etapa 2.3.2

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 $f (x) = \csc (x) \sec (x)$ e $g (x) = \tan (x)$ .

$- 5 (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) - \csc^{3} (x))) + 5 (\csc (x) \sec (x) \frac{d}{d x} [\tan (x)] + \tan (x) \frac{d}{d x} [\csc (x) \sec (x)])$

Etapa 2.3.3

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

$- 5 (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) - \csc^{3} (x))) + 5 (\csc (x) \sec (x) \sec^{2} (x) + \tan (x) \frac{d}{d x} [\csc (x) \sec (x)])$

Etapa 2.3.4

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

$- 5 (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) - \csc^{3} (x))) + 5 (\csc (x) \sec (x) \sec^{2} (x) + \tan (x) (\csc (x) \frac{d}{d x} [\sec (x)] + \sec (x) \frac{d}{d x} [\csc (x)]))$

Etapa 2.3.5

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

$- 5 (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) - \csc^{3} (x))) + 5 (\csc (x) \sec (x) \sec^{2} (x) + \tan (x) (\csc (x) (\sec (x) \tan (x)) + \sec (x) \frac{d}{d x} [\csc (x)]))$

Etapa 2.3.6

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

$- 5 (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) - \csc^{3} (x))) + 5 (\csc (x) \sec (x) \sec^{2} (x) + \tan (x) (\csc (x) (\sec (x) \tan (x)) + \sec (x) (- \csc (x) \cot (x))))$

Etapa 2.3.7

Multiplique $\sec (x)$ por $\sec^{2} (x)$ somando os expoentes.

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Etapa 2.3.7.1

Mova $\sec^{2} (x)$ .

$- 5 (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) - \csc^{3} (x))) + 5 (\csc (x) (\sec^{2} (x) \sec (x)) + \tan (x) (\csc (x) (\sec (x) \tan (x)) + \sec (x) (- \csc (x) \cot (x))))$

Etapa 2.3.7.2

Multiplique $\sec^{2} (x)$ por $\sec (x)$ .

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Etapa 2.3.7.2.1

Eleve $\sec (x)$ à potência de $1$ .

$- 5 (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) - \csc^{3} (x))) + 5 (\csc (x) (\sec^{2} (x) \sec^{1} (x)) + \tan (x) (\csc (x) (\sec (x) \tan (x)) + \sec (x) (- \csc (x) \cot (x))))$

Etapa 2.3.7.2.2

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

$- 5 (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) - \csc^{3} (x))) + 5 (\csc (x) {\sec (x)}^{2 + 1} + \tan (x) (\csc (x) (\sec (x) \tan (x)) + \sec (x) (- \csc (x) \cot (x))))$

Etapa 2.3.7.3

Some $2$ e $1$ .

$- 5 (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) - \csc^{3} (x))) + 5 (\csc (x) \sec^{3} (x) + \tan (x) (\csc (x) (\sec (x) \tan (x)) + \sec (x) (- \csc (x) \cot (x))))$

$- 5 (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x) - \csc^{3} (x))) + 5 (\csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot (x))))$

Etapa 2.4

Simplifique.

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Etapa 2.4.1

Aplique a propriedade distributiva.

$- 5 (\cot (x) \csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot^{2} (x)) + \sec (x) (- \csc^{3} (x))) + 5 (\csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot (x))))$

Etapa 2.4.2

Aplique a propriedade distributiva.

$- 5 (\cot (x) \csc (x) \sec (x) \tan (x)) - 5 (\sec (x) (- \csc (x) \cot^{2} (x))) - 5 (\sec (x) (- \csc^{3} (x))) + 5 (\csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x) + \sec (x) (- \csc (x) \cot (x))))$

Etapa 2.4.3

Aplique a propriedade distributiva.

$- 5 (\cot (x) \csc (x) \sec (x) \tan (x)) - 5 (\sec (x) (- \csc (x) \cot^{2} (x))) - 5 (\sec (x) (- \csc^{3} (x))) + 5 (\csc (x) \sec^{3} (x) + \tan (x) (\csc (x) \sec (x) \tan (x)) + \tan (x) (\sec (x) (- \csc (x) \cot (x))))$

Etapa 2.4.4

Aplique a propriedade distributiva.

$- 5 (\cot (x) \csc (x) \sec (x) \tan (x)) - 5 (\sec (x) (- \csc (x) \cot^{2} (x))) - 5 (\sec (x) (- \csc^{3} (x))) + 5 (\csc (x) \sec^{3} (x)) + 5 (\tan (x) (\csc (x) \sec (x) \tan (x))) + 5 (\tan (x) (\sec (x) (- \csc (x) \cot (x))))$

Etapa 2.4.5

Combine os termos.

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Etapa 2.4.5.1

Multiplique $- 1$ por $- 5$ .

$- 5 \cot (x) \csc (x) \sec (x) \tan (x) + 5 (\sec (x) (\csc (x) \cot^{2} (x))) - 5 (\sec (x) (- \csc^{3} (x))) + 5 (\csc (x) \sec^{3} (x)) + 5 (\tan (x) (\csc (x) \sec (x) \tan (x))) + 5 (\tan (x) (\sec (x) (- \csc (x) \cot (x))))$

Etapa 2.4.5.2

Multiplique $- 1$ por $- 5$ .

$- 5 \cot (x) \csc (x) \sec (x) \tan (x) + 5 \sec (x) \csc (x) \cot^{2} (x) + 5 (\sec (x) (\csc^{3} (x))) + 5 (\csc (x) \sec^{3} (x)) + 5 (\tan (x) (\csc (x) \sec (x) \tan (x))) + 5 (\tan (x) (\sec (x) (- \csc (x) \cot (x))))$

Etapa 2.4.5.3

Eleve $\tan (x)$ à potência de $1$ .

$- 5 \cot (x) \csc (x) \sec (x) \tan (x) + 5 \sec (x) \csc (x) \cot^{2} (x) + 5 \sec (x) \csc^{3} (x) + 5 \csc (x) \sec^{3} (x) + 5 (\tan^{1} (x) \tan (x) (\csc (x) \sec (x))) + 5 (\tan (x) (\sec (x) (- \csc (x) \cot (x))))$

Etapa 2.4.5.4

Eleve $\tan (x)$ à potência de $1$ .

$- 5 \cot (x) \csc (x) \sec (x) \tan (x) + 5 \sec (x) \csc (x) \cot^{2} (x) + 5 \sec (x) \csc^{3} (x) + 5 \csc (x) \sec^{3} (x) + 5 (\tan^{1} (x) \tan^{1} (x) (\csc (x) \sec (x))) + 5 (\tan (x) (\sec (x) (- \csc (x) \cot (x))))$

Etapa 2.4.5.5

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

$- 5 \cot (x) \csc (x) \sec (x) \tan (x) + 5 \sec (x) \csc (x) \cot^{2} (x) + 5 \sec (x) \csc^{3} (x) + 5 \csc (x) \sec^{3} (x) + 5 ({\tan (x)}^{1 + 1} (\csc (x) \sec (x))) + 5 (\tan (x) (\sec (x) (- \csc (x) \cot (x))))$

Etapa 2.4.5.6

Some $1$ e $1$ .

$- 5 \cot (x) \csc (x) \sec (x) \tan (x) + 5 \sec (x) \csc (x) \cot^{2} (x) + 5 \sec (x) \csc^{3} (x) + 5 \csc (x) \sec^{3} (x) + 5 (\tan^{2} (x) (\csc (x) \sec (x))) + 5 (\tan (x) (\sec (x) (- \csc (x) \cot (x))))$

Etapa 2.4.5.7

Multiplique $- 1$ por $5$ .

$- 5 \cot (x) \csc (x) \sec (x) \tan (x) + 5 \sec (x) \csc (x) \cot^{2} (x) + 5 \sec (x) \csc^{3} (x) + 5 \csc (x) \sec^{3} (x) + 5 \tan^{2} (x) \csc (x) \sec (x) - 5 (\tan (x) \sec (x) (\csc (x) \cot (x)))$

Etapa 2.4.5.8

Reordene os fatores de $- 5 \tan (x) \sec (x) \csc (x) \cot (x)$ .

$- 5 \cot (x) \csc (x) \sec (x) \tan (x) + 5 \sec (x) \csc (x) \cot^{2} (x) + 5 \sec (x) \csc^{3} (x) + 5 \csc (x) \sec^{3} (x) + 5 \tan^{2} (x) \csc (x) \sec (x) - 5 \cot (x) \csc (x) \sec (x) \tan (x)$

Etapa 2.4.5.9

Subtraia $5 \cot (x) \csc (x) \sec (x) \tan (x)$ de $- 5 \cot (x) \csc (x) \sec (x) \tan (x)$ .

$- 10 \cot (x) \csc (x) \sec (x) \tan (x) + 5 \sec (x) \csc (x) \cot^{2} (x) + 5 \sec (x) \csc^{3} (x) + 5 \csc (x) \sec^{3} (x) + 5 \tan^{2} (x) \csc (x) \sec (x)$

Etapa 2.4.6

Reordene os termos.

$f'' (x) = - 10 \cot (x) \csc (x) \sec (x) \tan (x) + 5 \cot^{2} (x) \csc (x) \sec (x) + 5 \csc^{3} (x) \sec (x) + 5 \sec^{3} (x) \csc (x) + 5 \tan^{2} (x) \csc (x) \sec (x)$