Cálculo Exemplos

$y = \sin (3 x^{2} e^{x})$

Etapa 1

Encontre a primeira derivada.

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Etapa 1.1

Diferencie usando a regra da cadeia, que determina que $\frac{d}{d x} [f (g (x))]$ é $f' (g (x)) g' (x)$ , em que $f (x) = \sin (x)$ e $g (x) = 3 x^{2} e^{x}$ .

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Etapa 1.1.1

Para aplicar a regra da cadeia, defina $u$ como $3 x^{2} e^{x}$ .

$\frac{d}{d u} [\sin (u)] \frac{d}{d x} [3 x^{2} e^{x}]$

Etapa 1.1.2

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

$\cos (u) \frac{d}{d x} [3 x^{2} e^{x}]$

Etapa 1.1.3

Substitua todas as ocorrências de $u$ por $3 x^{2} e^{x}$ .

$\cos (3 x^{2} e^{x}) \frac{d}{d x} [3 x^{2} e^{x}]$

Etapa 1.2

Como $3$ é constante em relação a $x$ , a derivada de $3 x^{2} e^{x}$ em relação a $x$ é $3 \frac{d}{d x} [x^{2} e^{x}]$ .

$\cos (3 x^{2} e^{x}) (3 \frac{d}{d x} [x^{2} e^{x}])$

Etapa 1.3

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 $f (x) = x^{2}$ e $g (x) = e^{x}$ .

$\cos (3 x^{2} e^{x}) (3 (x^{2} \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{2}]))$

Etapa 1.4

Diferencie usando a regra exponencial, que determina que $\frac{d}{d x} [a^{x}]$ é $a^{x} \ln (a)$ , em que $a$ = $e$ .

$\cos (3 x^{2} e^{x}) (3 (x^{2} e^{x} + e^{x} \frac{d}{d x} [x^{2}]))$

Etapa 1.5

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

$\cos (3 x^{2} e^{x}) (3 (x^{2} e^{x} + e^{x} (2 x)))$

Etapa 1.6

Simplifique.

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Etapa 1.6.1

Aplique a propriedade distributiva.

$\cos (3 x^{2} e^{x}) (3 (x^{2} e^{x}) + 3 (e^{x} (2 x)))$

Etapa 1.6.2

Aplique a propriedade distributiva.

$\cos (3 x^{2} e^{x}) (3 (x^{2} e^{x})) + \cos (3 x^{2} e^{x}) (3 (e^{x} (2 x)))$

Etapa 1.6.3

Multiplique $2$ por $3$ .

$\cos (3 x^{2} e^{x}) (3 x^{2} e^{x}) + \cos (3 x^{2} e^{x}) (6 e^{x} x)$

Etapa 1.6.4

Reordene os termos.

$f' (x) = 3 e^{x} x^{2} \cos (3 x^{2} e^{x}) + 6 e^{x} x \cos (3 x^{2} e^{x})$

Etapa 2

Encontre a segunda derivada.

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Etapa 2.1

De acordo com a regra da soma, a derivada de $3 e^{x} x^{2} \cos (3 x^{2} e^{x}) + 6 e^{x} x \cos (3 x^{2} e^{x})$ com relação a $x$ é $\frac{d}{d x} [3 e^{x} x^{2} \cos (3 x^{2} e^{x})] + \frac{d}{d x} [6 e^{x} x \cos (3 x^{2} e^{x})]$ .

$\frac{d}{d x} [3 e^{x} x^{2} \cos (3 x^{2} e^{x})] + \frac{d}{d x} [6 e^{x} x \cos (3 x^{2} e^{x})]$

Etapa 2.2

Avalie $\frac{d}{d x} [3 e^{x} x^{2} \cos (3 x^{2} e^{x})]$ .

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Etapa 2.2.1

Como $3$ é constante em relação a $x$ , a derivada de $3 e^{x} x^{2} \cos (3 x^{2} e^{x})$ em relação a $x$ é $3 \frac{d}{d x} [e^{x} x^{2} \cos (3 x^{2} e^{x})]$ .

$3 \frac{d}{d x} [e^{x} x^{2} \cos (3 x^{2} e^{x})] + \frac{d}{d x} [6 e^{x} x \cos (3 x^{2} e^{x})]$

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 $f (x) = e^{x} x^{2}$ e $g (x) = \cos (3 x^{2} e^{x})$ .

$3 (e^{x} x^{2} \frac{d}{d x} [\cos (3 x^{2} e^{x})] + \cos (3 x^{2} e^{x}) \frac{d}{d x} [e^{x} x^{2}]) + \frac{d}{d x} [6 e^{x} x \cos (3 x^{2} e^{x})]$

Etapa 2.2.3

Diferencie usando a regra da cadeia, que determina que $\frac{d}{d x} [f (g (x))]$ é $f' (g (x)) g' (x)$ , em que $f (x) = \cos (x)$ e $g (x) = 3 x^{2} e^{x}$ .

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Etapa 2.2.3.1

Para aplicar a regra da cadeia, defina $u_{1}$ como $3 x^{2} e^{x}$ .

$3 (e^{x} x^{2} (\frac{d}{d u_{1}} [\cos (u_{1})] \frac{d}{d x} [3 x^{2} e^{x}]) + \cos (3 x^{2} e^{x}) \frac{d}{d x} [e^{x} x^{2}]) + \frac{d}{d x} [6 e^{x} x \cos (3 x^{2} e^{x})]$

Etapa 2.2.3.2

A derivada de $\cos (u_{1})$ em relação a $u_{1}$ é $- \sin (u_{1})$ .

$3 (e^{x} x^{2} (- \sin (u_{1}) \frac{d}{d x} [3 x^{2} e^{x}]) + \cos (3 x^{2} e^{x}) \frac{d}{d x} [e^{x} x^{2}]) + \frac{d}{d x} [6 e^{x} x \cos (3 x^{2} e^{x})]$

Etapa 2.2.3.3

Substitua todas as ocorrências de $u_{1}$ por $3 x^{2} e^{x}$ .

$3 (e^{x} x^{2} (- \sin (3 x^{2} e^{x}) \frac{d}{d x} [3 x^{2} e^{x}]) + \cos (3 x^{2} e^{x}) \frac{d}{d x} [e^{x} x^{2}]) + \frac{d}{d x} [6 e^{x} x \cos (3 x^{2} e^{x})]$

Etapa 2.2.4

Como $3$ é constante em relação a $x$ , a derivada de $3 x^{2} e^{x}$ em relação a $x$ é $3 \frac{d}{d x} [x^{2} e^{x}]$ .

$3 (e^{x} x^{2} (- \sin (3 x^{2} e^{x}) (3 \frac{d}{d x} [x^{2} e^{x}])) + \cos (3 x^{2} e^{x}) \frac{d}{d x} [e^{x} x^{2}]) + \frac{d}{d x} [6 e^{x} x \cos (3 x^{2} e^{x})]$

Etapa 2.2.5

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 $f (x) = x^{2}$ e $g (x) = e^{x}$ .

$3 (e^{x} x^{2} (- \sin (3 x^{2} e^{x}) (3 (x^{2} \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{2}]))) + \cos (3 x^{2} e^{x}) \frac{d}{d x} [e^{x} x^{2}]) + \frac{d}{d x} [6 e^{x} x \cos (3 x^{2} e^{x})]$

Etapa 2.2.6

Diferencie usando a regra exponencial, que determina que $\frac{d}{d x} [a^{x}]$ é $a^{x} \ln (a)$ , em que $a$ = $e$ .

$3 (e^{x} x^{2} (- \sin (3 x^{2} e^{x}) (3 (x^{2} e^{x} + e^{x} \frac{d}{d x} [x^{2}]))) + \cos (3 x^{2} e^{x}) \frac{d}{d x} [e^{x} x^{2}]) + \frac{d}{d x} [6 e^{x} x \cos (3 x^{2} e^{x})]$

Etapa 2.2.7

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

$3 (e^{x} x^{2} (- \sin (3 x^{2} e^{x}) (3 (x^{2} e^{x} + e^{x} (2 x)))) + \cos (3 x^{2} e^{x}) \frac{d}{d x} [e^{x} x^{2}]) + \frac{d}{d x} [6 e^{x} x \cos (3 x^{2} e^{x})]$

Etapa 2.2.8

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 $f (x) = e^{x}$ e $g (x) = x^{2}$ .

$3 (e^{x} x^{2} (- \sin (3 x^{2} e^{x}) (3 (x^{2} e^{x} + e^{x} (2 x)))) + \cos (3 x^{2} e^{x}) (e^{x} \frac{d}{d x} [x^{2}] + x^{2} \frac{d}{d x} [e^{x}])) + \frac{d}{d x} [6 e^{x} x \cos (3 x^{2} e^{x})]$

Etapa 2.2.9

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

$3 (e^{x} x^{2} (- \sin (3 x^{2} e^{x}) (3 (x^{2} e^{x} + e^{x} (2 x)))) + \cos (3 x^{2} e^{x}) (e^{x} (2 x) + x^{2} \frac{d}{d x} [e^{x}])) + \frac{d}{d x} [6 e^{x} x \cos (3 x^{2} e^{x})]$

Etapa 2.2.10

Diferencie usando a regra exponencial, que determina que $\frac{d}{d x} [a^{x}]$ é $a^{x} \ln (a)$ , em que $a$ = $e$ .

$3 (e^{x} x^{2} (- \sin (3 x^{2} e^{x}) (3 (x^{2} e^{x} + e^{x} (2 x)))) + \cos (3 x^{2} e^{x}) (e^{x} (2 x) + x^{2} e^{x})) + \frac{d}{d x} [6 e^{x} x \cos (3 x^{2} e^{x})]$

Etapa 2.2.11

Multiplique $3$ por $- 1$ .

$3 (e^{x} x^{2} (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x} + e^{x} (2 x))) + \cos (3 x^{2} e^{x}) (e^{x} (2 x) + x^{2} e^{x})) + \frac{d}{d x} [6 e^{x} x \cos (3 x^{2} e^{x})]$

Etapa 2.3

Avalie $\frac{d}{d x} [6 e^{x} x \cos (3 x^{2} e^{x})]$ .

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Etapa 2.3.1

Como $6$ é constante em relação a $x$ , a derivada de $6 e^{x} x \cos (3 x^{2} e^{x})$ em relação a $x$ é $6 \frac{d}{d x} [e^{x} x \cos (3 x^{2} e^{x})]$ .

$3 (e^{x} x^{2} (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x} + e^{x} (2 x))) + \cos (3 x^{2} e^{x}) (e^{x} (2 x) + x^{2} e^{x})) + 6 \frac{d}{d x} [e^{x} x \cos (3 x^{2} e^{x})]$

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 $f (x) = e^{x} x$ e $g (x) = \cos (3 x^{2} e^{x})$ .

$3 (e^{x} x^{2} (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x} + e^{x} (2 x))) + \cos (3 x^{2} e^{x}) (e^{x} (2 x) + x^{2} e^{x})) + 6 (e^{x} x \frac{d}{d x} [\cos (3 x^{2} e^{x})] + \cos (3 x^{2} e^{x}) \frac{d}{d x} [e^{x} x])$

Etapa 2.3.3

Diferencie usando a regra da cadeia, que determina que $\frac{d}{d x} [f (g (x))]$ é $f' (g (x)) g' (x)$ , em que $f (x) = \cos (x)$ e $g (x) = 3 x^{2} e^{x}$ .

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Etapa 2.3.3.1

Para aplicar a regra da cadeia, defina $u_{2}$ como $3 x^{2} e^{x}$ .

$3 (e^{x} x^{2} (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x} + e^{x} (2 x))) + \cos (3 x^{2} e^{x}) (e^{x} (2 x) + x^{2} e^{x})) + 6 (e^{x} x (\frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d x} [3 x^{2} e^{x}]) + \cos (3 x^{2} e^{x}) \frac{d}{d x} [e^{x} x])$

Etapa 2.3.3.2

A derivada de $\cos (u_{2})$ em relação a $u_{2}$ é $- \sin (u_{2})$ .

$3 (e^{x} x^{2} (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x} + e^{x} (2 x))) + \cos (3 x^{2} e^{x}) (e^{x} (2 x) + x^{2} e^{x})) + 6 (e^{x} x (- \sin (u_{2}) \frac{d}{d x} [3 x^{2} e^{x}]) + \cos (3 x^{2} e^{x}) \frac{d}{d x} [e^{x} x])$

Etapa 2.3.3.3

Substitua todas as ocorrências de $u_{2}$ por $3 x^{2} e^{x}$ .

$3 (e^{x} x^{2} (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x} + e^{x} (2 x))) + \cos (3 x^{2} e^{x}) (e^{x} (2 x) + x^{2} e^{x})) + 6 (e^{x} x (- \sin (3 x^{2} e^{x}) \frac{d}{d x} [3 x^{2} e^{x}]) + \cos (3 x^{2} e^{x}) \frac{d}{d x} [e^{x} x])$

Etapa 2.3.4

Como $3$ é constante em relação a $x$ , a derivada de $3 x^{2} e^{x}$ em relação a $x$ é $3 \frac{d}{d x} [x^{2} e^{x}]$ .

$3 (e^{x} x^{2} (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x} + e^{x} (2 x))) + \cos (3 x^{2} e^{x}) (e^{x} (2 x) + x^{2} e^{x})) + 6 (e^{x} x (- \sin (3 x^{2} e^{x}) (3 \frac{d}{d x} [x^{2} e^{x}])) + \cos (3 x^{2} e^{x}) \frac{d}{d x} [e^{x} x])$

Etapa 2.3.5

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 $f (x) = x^{2}$ e $g (x) = e^{x}$ .

$3 (e^{x} x^{2} (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x} + e^{x} (2 x))) + \cos (3 x^{2} e^{x}) (e^{x} (2 x) + x^{2} e^{x})) + 6 (e^{x} x (- \sin (3 x^{2} e^{x}) (3 (x^{2} \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{2}]))) + \cos (3 x^{2} e^{x}) \frac{d}{d x} [e^{x} x])$

Etapa 2.3.6

Diferencie usando a regra exponencial, que determina que $\frac{d}{d x} [a^{x}]$ é $a^{x} \ln (a)$ , em que $a$ = $e$ .

$3 (e^{x} x^{2} (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x} + e^{x} (2 x))) + \cos (3 x^{2} e^{x}) (e^{x} (2 x) + x^{2} e^{x})) + 6 (e^{x} x (- \sin (3 x^{2} e^{x}) (3 (x^{2} e^{x} + e^{x} \frac{d}{d x} [x^{2}]))) + \cos (3 x^{2} e^{x}) \frac{d}{d x} [e^{x} x])$

Etapa 2.3.7

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

$3 (e^{x} x^{2} (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x} + e^{x} (2 x))) + \cos (3 x^{2} e^{x}) (e^{x} (2 x) + x^{2} e^{x})) + 6 (e^{x} x (- \sin (3 x^{2} e^{x}) (3 (x^{2} e^{x} + e^{x} (2 x)))) + \cos (3 x^{2} e^{x}) \frac{d}{d x} [e^{x} x])$

Etapa 2.3.8

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 $f (x) = e^{x}$ e $g (x) = x$ .

Etapa 2.3.9

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

Etapa 2.3.10

Diferencie usando a regra exponencial, que determina que $\frac{d}{d x} [a^{x}]$ é $a^{x} \ln (a)$ , em que $a$ = $e$ .

Etapa 2.3.11

Multiplique $3$ por $- 1$ .

$3 (e^{x} x^{2} (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x} + e^{x} (2 x))) + \cos (3 x^{2} e^{x}) (e^{x} (2 x) + x^{2} e^{x})) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x} + e^{x} (2 x))) + \cos (3 x^{2} e^{x}) (e^{x} \cdot 1 + x e^{x}))$

Etapa 2.3.12

Multiplique $e^{x}$ por $1$ .

Etapa 2.4

Simplifique.

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Etapa 2.4.1

Aplique a propriedade distributiva.

$3 (e^{x} x^{2} (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x}) - 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x))) + \cos (3 x^{2} e^{x}) (e^{x} (2 x) + x^{2} e^{x})) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x} + e^{x} (2 x))) + \cos (3 x^{2} e^{x}) (e^{x} + x e^{x}))$

Etapa 2.4.2

Aplique a propriedade distributiva.

$3 (e^{x} x^{2} (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x})) + e^{x} x^{2} (- 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x))) + \cos (3 x^{2} e^{x}) (e^{x} (2 x) + x^{2} e^{x})) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x} + e^{x} (2 x))) + \cos (3 x^{2} e^{x}) (e^{x} + x e^{x}))$

Etapa 2.4.3

Aplique a propriedade distributiva.

$3 (e^{x} x^{2} (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x})) + e^{x} x^{2} (- 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x))) + \cos (3 x^{2} e^{x}) (e^{x} (2 x)) + \cos (3 x^{2} e^{x}) (x^{2} e^{x})) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x} + e^{x} (2 x))) + \cos (3 x^{2} e^{x}) (e^{x} + x e^{x}))$

Etapa 2.4.4

Aplique a propriedade distributiva.

$3 (e^{x} x^{2} (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x}))) + 3 (e^{x} x^{2} (- 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x)))) + 3 (\cos (3 x^{2} e^{x}) (e^{x} (2 x))) + 3 (\cos (3 x^{2} e^{x}) (x^{2} e^{x})) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x} + e^{x} (2 x))) + \cos (3 x^{2} e^{x}) (e^{x} + x e^{x}))$

Etapa 2.4.5

Aplique a propriedade distributiva.

$3 (e^{x} x^{2} (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x}))) + 3 (e^{x} x^{2} (- 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x)))) + 3 (\cos (3 x^{2} e^{x}) (e^{x} (2 x))) + 3 (\cos (3 x^{2} e^{x}) (x^{2} e^{x})) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x}) - 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x))) + \cos (3 x^{2} e^{x}) (e^{x} + x e^{x}))$

Etapa 2.4.6

Aplique a propriedade distributiva.

$3 (e^{x} x^{2} (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x}))) + 3 (e^{x} x^{2} (- 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x)))) + 3 (\cos (3 x^{2} e^{x}) (e^{x} (2 x))) + 3 (\cos (3 x^{2} e^{x}) (x^{2} e^{x})) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x})) + e^{x} x (- 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x))) + \cos (3 x^{2} e^{x}) (e^{x} + x e^{x}))$

Etapa 2.4.7

Aplique a propriedade distributiva.

$3 (e^{x} x^{2} (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x}))) + 3 (e^{x} x^{2} (- 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x)))) + 3 (\cos (3 x^{2} e^{x}) (e^{x} (2 x))) + 3 (\cos (3 x^{2} e^{x}) (x^{2} e^{x})) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x})) + e^{x} x (- 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x))) + \cos (3 x^{2} e^{x}) e^{x} + \cos (3 x^{2} e^{x}) (x e^{x}))$

Etapa 2.4.8

Aplique a propriedade distributiva.

$3 (e^{x} x^{2} (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x}))) + 3 (e^{x} x^{2} (- 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x)))) + 3 (\cos (3 x^{2} e^{x}) (e^{x} (2 x))) + 3 (\cos (3 x^{2} e^{x}) (x^{2} e^{x})) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x}))) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x)))) + 6 (\cos (3 x^{2} e^{x}) e^{x}) + 6 (\cos (3 x^{2} e^{x}) (x e^{x}))$

Etapa 2.4.9

Combine os termos.

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Etapa 2.4.9.1

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

$3 (x^{2} (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x + x}))) + 3 (e^{x} x^{2} (- 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x)))) + 3 (\cos (3 x^{2} e^{x}) (e^{x} (2 x))) + 3 (\cos (3 x^{2} e^{x}) (x^{2} e^{x})) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x}))) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x)))) + 6 (\cos (3 x^{2} e^{x}) e^{x}) + 6 (\cos (3 x^{2} e^{x}) (x e^{x}))$

Etapa 2.4.9.2

Some $x$ e $x$ .

$3 (x^{2} (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{2 x}))) + 3 (e^{x} x^{2} (- 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x)))) + 3 (\cos (3 x^{2} e^{x}) (e^{x} (2 x))) + 3 (\cos (3 x^{2} e^{x}) (x^{2} e^{x})) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x}))) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x)))) + 6 (\cos (3 x^{2} e^{x}) e^{x}) + 6 (\cos (3 x^{2} e^{x}) (x e^{x}))$

Etapa 2.4.9.3

Multiplique $x^{2}$ por $x^{2}$ somando os expoentes.

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Etapa 2.4.9.3.1

Mova $x^{2}$ .

$3 (x^{2} x^{2} (- 3 \sin (3 x^{2} e^{x}) (e^{2 x}))) + 3 (e^{x} x^{2} (- 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x)))) + 3 (\cos (3 x^{2} e^{x}) (e^{x} (2 x))) + 3 (\cos (3 x^{2} e^{x}) (x^{2} e^{x})) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x}))) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x)))) + 6 (\cos (3 x^{2} e^{x}) e^{x}) + 6 (\cos (3 x^{2} e^{x}) (x e^{x}))$

Etapa 2.4.9.3.2

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

$3 (x^{2 + 2} (- 3 \sin (3 x^{2} e^{x}) (e^{2 x}))) + 3 (e^{x} x^{2} (- 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x)))) + 3 (\cos (3 x^{2} e^{x}) (e^{x} (2 x))) + 3 (\cos (3 x^{2} e^{x}) (x^{2} e^{x})) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x}))) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x)))) + 6 (\cos (3 x^{2} e^{x}) e^{x}) + 6 (\cos (3 x^{2} e^{x}) (x e^{x}))$

Etapa 2.4.9.3.3

Some $2$ e $2$ .

$3 (x^{4} (- 3 \sin (3 x^{2} e^{x}) (e^{2 x}))) + 3 (e^{x} x^{2} (- 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x)))) + 3 (\cos (3 x^{2} e^{x}) (e^{x} (2 x))) + 3 (\cos (3 x^{2} e^{x}) (x^{2} e^{x})) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x}))) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x)))) + 6 (\cos (3 x^{2} e^{x}) e^{x}) + 6 (\cos (3 x^{2} e^{x}) (x e^{x}))$

Etapa 2.4.9.4

Multiplique $- 3$ por $3$ .

$- 9 (x^{4} (\sin (3 x^{2} e^{x}) (e^{2 x}))) + 3 (e^{x} x^{2} (- 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x)))) + 3 (\cos (3 x^{2} e^{x}) (e^{x} (2 x))) + 3 (\cos (3 x^{2} e^{x}) (x^{2} e^{x})) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x}))) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x)))) + 6 (\cos (3 x^{2} e^{x}) e^{x}) + 6 (\cos (3 x^{2} e^{x}) (x e^{x}))$

Etapa 2.4.9.5

Multiplique $2$ por $- 3$ .

$- 9 x^{4} (\sin (3 x^{2} e^{x}) (e^{2 x})) + 3 (e^{x} x^{2} (- 6 \sin (3 x^{2} e^{x}) (e^{x} (x)))) + 3 (\cos (3 x^{2} e^{x}) (e^{x} (2 x))) + 3 (\cos (3 x^{2} e^{x}) (x^{2} e^{x})) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x}))) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x)))) + 6 (\cos (3 x^{2} e^{x}) e^{x}) + 6 (\cos (3 x^{2} e^{x}) (x e^{x}))$

Etapa 2.4.9.6

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

$- 9 x^{4} (\sin (3 x^{2} e^{x}) (e^{2 x})) + 3 (x^{2} (- 6 \sin (3 x^{2} e^{x}) (e^{x + x} x))) + 3 (\cos (3 x^{2} e^{x}) (e^{x} (2 x))) + 3 (\cos (3 x^{2} e^{x}) (x^{2} e^{x})) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x}))) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x)))) + 6 (\cos (3 x^{2} e^{x}) e^{x}) + 6 (\cos (3 x^{2} e^{x}) (x e^{x}))$

Etapa 2.4.9.7

Some $x$ e $x$ .

$- 9 x^{4} (\sin (3 x^{2} e^{x}) (e^{2 x})) + 3 (x^{2} (- 6 \sin (3 x^{2} e^{x}) (e^{2 x} x))) + 3 (\cos (3 x^{2} e^{x}) (e^{x} (2 x))) + 3 (\cos (3 x^{2} e^{x}) (x^{2} e^{x})) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x}))) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x)))) + 6 (\cos (3 x^{2} e^{x}) e^{x}) + 6 (\cos (3 x^{2} e^{x}) (x e^{x}))$

Etapa 2.4.9.8

Multiplique $x^{2}$ por $x$ somando os expoentes.

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Etapa 2.4.9.8.1

Mova $x$ .

$- 9 x^{4} (\sin (3 x^{2} e^{x}) (e^{2 x})) + 3 (x \cdot x^{2} (- 6 \sin (3 x^{2} e^{x}) (e^{2 x}))) + 3 (\cos (3 x^{2} e^{x}) (e^{x} (2 x))) + 3 (\cos (3 x^{2} e^{x}) (x^{2} e^{x})) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x}))) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x)))) + 6 (\cos (3 x^{2} e^{x}) e^{x}) + 6 (\cos (3 x^{2} e^{x}) (x e^{x}))$

Etapa 2.4.9.8.2

Multiplique $x$ por $x^{2}$ .

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Etapa 2.4.9.8.2.1

Eleve $x$ à potência de $1$ .

$- 9 x^{4} (\sin (3 x^{2} e^{x}) (e^{2 x})) + 3 (x^{1} x^{2} (- 6 \sin (3 x^{2} e^{x}) (e^{2 x}))) + 3 (\cos (3 x^{2} e^{x}) (e^{x} (2 x))) + 3 (\cos (3 x^{2} e^{x}) (x^{2} e^{x})) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x}))) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x)))) + 6 (\cos (3 x^{2} e^{x}) e^{x}) + 6 (\cos (3 x^{2} e^{x}) (x e^{x}))$

Etapa 2.4.9.8.2.2

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

$- 9 x^{4} (\sin (3 x^{2} e^{x}) (e^{2 x})) + 3 (x^{1 + 2} (- 6 \sin (3 x^{2} e^{x}) (e^{2 x}))) + 3 (\cos (3 x^{2} e^{x}) (e^{x} (2 x))) + 3 (\cos (3 x^{2} e^{x}) (x^{2} e^{x})) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x}))) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x)))) + 6 (\cos (3 x^{2} e^{x}) e^{x}) + 6 (\cos (3 x^{2} e^{x}) (x e^{x}))$

Etapa 2.4.9.8.3

Some $1$ e $2$ .

$- 9 x^{4} (\sin (3 x^{2} e^{x}) (e^{2 x})) + 3 (x^{3} (- 6 \sin (3 x^{2} e^{x}) (e^{2 x}))) + 3 (\cos (3 x^{2} e^{x}) (e^{x} (2 x))) + 3 (\cos (3 x^{2} e^{x}) (x^{2} e^{x})) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x}))) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x)))) + 6 (\cos (3 x^{2} e^{x}) e^{x}) + 6 (\cos (3 x^{2} e^{x}) (x e^{x}))$

Etapa 2.4.9.9

Multiplique $- 6$ por $3$ .

$- 9 x^{4} (\sin (3 x^{2} e^{x}) (e^{2 x})) - 18 (x^{3} (\sin (3 x^{2} e^{x}) (e^{2 x}))) + 3 (\cos (3 x^{2} e^{x}) (e^{x} (2 x))) + 3 (\cos (3 x^{2} e^{x}) (x^{2} e^{x})) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x}))) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x)))) + 6 (\cos (3 x^{2} e^{x}) e^{x}) + 6 (\cos (3 x^{2} e^{x}) (x e^{x}))$

Etapa 2.4.9.10

Multiplique $2$ por $3$ .

$- 9 x^{4} (\sin (3 x^{2} e^{x}) (e^{2 x})) - 18 x^{3} (\sin (3 x^{2} e^{x}) (e^{2 x})) + 6 (\cos (3 x^{2} e^{x}) (e^{x} (x))) + 3 (\cos (3 x^{2} e^{x}) (x^{2} e^{x})) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x}))) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x)))) + 6 (\cos (3 x^{2} e^{x}) e^{x}) + 6 (\cos (3 x^{2} e^{x}) (x e^{x}))$

Etapa 2.4.9.11

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

$- 9 x^{4} (\sin (3 x^{2} e^{x}) (e^{2 x})) - 18 x^{3} (\sin (3 x^{2} e^{x}) (e^{2 x})) + 6 \cos (3 x^{2} e^{x}) (e^{x} (x)) + 3 \cos (3 x^{2} e^{x}) (x^{2} e^{x}) + 6 (x (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{x + x}))) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x)))) + 6 (\cos (3 x^{2} e^{x}) e^{x}) + 6 (\cos (3 x^{2} e^{x}) (x e^{x}))$

Etapa 2.4.9.12

Some $x$ e $x$ .

$- 9 x^{4} (\sin (3 x^{2} e^{x}) (e^{2 x})) - 18 x^{3} (\sin (3 x^{2} e^{x}) (e^{2 x})) + 6 \cos (3 x^{2} e^{x}) (e^{x} (x)) + 3 \cos (3 x^{2} e^{x}) (x^{2} e^{x}) + 6 (x (- 3 \sin (3 x^{2} e^{x}) (x^{2} e^{2 x}))) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x)))) + 6 (\cos (3 x^{2} e^{x}) e^{x}) + 6 (\cos (3 x^{2} e^{x}) (x e^{x}))$

Etapa 2.4.9.13

Eleve $x$ à potência de $1$ .

$- 9 x^{4} (\sin (3 x^{2} e^{x}) (e^{2 x})) - 18 x^{3} (\sin (3 x^{2} e^{x}) (e^{2 x})) + 6 \cos (3 x^{2} e^{x}) (e^{x} (x)) + 3 \cos (3 x^{2} e^{x}) (x^{2} e^{x}) + 6 (x^{2} x^{1} (- 3 \sin (3 x^{2} e^{x}) (e^{2 x}))) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x)))) + 6 (\cos (3 x^{2} e^{x}) e^{x}) + 6 (\cos (3 x^{2} e^{x}) (x e^{x}))$

Etapa 2.4.9.14

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

$- 9 x^{4} (\sin (3 x^{2} e^{x}) (e^{2 x})) - 18 x^{3} (\sin (3 x^{2} e^{x}) (e^{2 x})) + 6 \cos (3 x^{2} e^{x}) (e^{x} (x)) + 3 \cos (3 x^{2} e^{x}) (x^{2} e^{x}) + 6 (x^{2 + 1} (- 3 \sin (3 x^{2} e^{x}) (e^{2 x}))) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x)))) + 6 (\cos (3 x^{2} e^{x}) e^{x}) + 6 (\cos (3 x^{2} e^{x}) (x e^{x}))$

Etapa 2.4.9.15

Some $2$ e $1$ .

$- 9 x^{4} (\sin (3 x^{2} e^{x}) (e^{2 x})) - 18 x^{3} (\sin (3 x^{2} e^{x}) (e^{2 x})) + 6 \cos (3 x^{2} e^{x}) (e^{x} (x)) + 3 \cos (3 x^{2} e^{x}) (x^{2} e^{x}) + 6 (x^{3} (- 3 \sin (3 x^{2} e^{x}) (e^{2 x}))) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x)))) + 6 (\cos (3 x^{2} e^{x}) e^{x}) + 6 (\cos (3 x^{2} e^{x}) (x e^{x}))$

Etapa 2.4.9.16

Multiplique $- 3$ por $6$ .

$- 9 x^{4} (\sin (3 x^{2} e^{x}) (e^{2 x})) - 18 x^{3} (\sin (3 x^{2} e^{x}) (e^{2 x})) + 6 \cos (3 x^{2} e^{x}) (e^{x} (x)) + 3 \cos (3 x^{2} e^{x}) (x^{2} e^{x}) - 18 (x^{3} (\sin (3 x^{2} e^{x}) (e^{2 x}))) + 6 (e^{x} x (- 3 \sin (3 x^{2} e^{x}) (e^{x} (2 x)))) + 6 (\cos (3 x^{2} e^{x}) e^{x}) + 6 (\cos (3 x^{2} e^{x}) (x e^{x}))$

Etapa 2.4.9.17

Multiplique $2$ por $- 3$ .

$- 9 x^{4} (\sin (3 x^{2} e^{x}) (e^{2 x})) - 18 x^{3} (\sin (3 x^{2} e^{x}) (e^{2 x})) + 6 \cos (3 x^{2} e^{x}) (e^{x} (x)) + 3 \cos (3 x^{2} e^{x}) (x^{2} e^{x}) - 18 x^{3} (\sin (3 x^{2} e^{x}) (e^{2 x})) + 6 (e^{x} x (- 6 \sin (3 x^{2} e^{x}) (e^{x} (x)))) + 6 (\cos (3 x^{2} e^{x}) e^{x}) + 6 (\cos (3 x^{2} e^{x}) (x e^{x}))$

Etapa 2.4.9.18

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

$- 9 x^{4} (\sin (3 x^{2} e^{x}) (e^{2 x})) - 18 x^{3} (\sin (3 x^{2} e^{x}) (e^{2 x})) + 6 \cos (3 x^{2} e^{x}) (e^{x} (x)) + 3 \cos (3 x^{2} e^{x}) (x^{2} e^{x}) - 18 x^{3} (\sin (3 x^{2} e^{x}) (e^{2 x})) + 6 (x (- 6 \sin (3 x^{2} e^{x}) (e^{x + x} x))) + 6 (\cos (3 x^{2} e^{x}) e^{x}) + 6 (\cos (3 x^{2} e^{x}) (x e^{x}))$

Etapa 2.4.9.19

Some $x$ e $x$ .

$- 9 x^{4} (\sin (3 x^{2} e^{x}) (e^{2 x})) - 18 x^{3} (\sin (3 x^{2} e^{x}) (e^{2 x})) + 6 \cos (3 x^{2} e^{x}) (e^{x} (x)) + 3 \cos (3 x^{2} e^{x}) (x^{2} e^{x}) - 18 x^{3} (\sin (3 x^{2} e^{x}) (e^{2 x})) + 6 (x (- 6 \sin (3 x^{2} e^{x}) (e^{2 x} x))) + 6 (\cos (3 x^{2} e^{x}) e^{x}) + 6 (\cos (3 x^{2} e^{x}) (x e^{x}))$

Etapa 2.4.9.20

Eleve $x$ à potência de $1$ .

$- 9 x^{4} (\sin (3 x^{2} e^{x}) (e^{2 x})) - 18 x^{3} (\sin (3 x^{2} e^{x}) (e^{2 x})) + 6 \cos (3 x^{2} e^{x}) (e^{x} (x)) + 3 \cos (3 x^{2} e^{x}) (x^{2} e^{x}) - 18 x^{3} (\sin (3 x^{2} e^{x}) (e^{2 x})) + 6 (x^{1} x (- 6 \sin (3 x^{2} e^{x}) (e^{2 x}))) + 6 (\cos (3 x^{2} e^{x}) e^{x}) + 6 (\cos (3 x^{2} e^{x}) (x e^{x}))$

Etapa 2.4.9.21

Eleve $x$ à potência de $1$ .

$- 9 x^{4} (\sin (3 x^{2} e^{x}) (e^{2 x})) - 18 x^{3} (\sin (3 x^{2} e^{x}) (e^{2 x})) + 6 \cos (3 x^{2} e^{x}) (e^{x} (x)) + 3 \cos (3 x^{2} e^{x}) (x^{2} e^{x}) - 18 x^{3} (\sin (3 x^{2} e^{x}) (e^{2 x})) + 6 (x^{1} x^{1} (- 6 \sin (3 x^{2} e^{x}) (e^{2 x}))) + 6 (\cos (3 x^{2} e^{x}) e^{x}) + 6 (\cos (3 x^{2} e^{x}) (x e^{x}))$

Etapa 2.4.9.22

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

$- 9 x^{4} (\sin (3 x^{2} e^{x}) (e^{2 x})) - 18 x^{3} (\sin (3 x^{2} e^{x}) (e^{2 x})) + 6 \cos (3 x^{2} e^{x}) (e^{x} (x)) + 3 \cos (3 x^{2} e^{x}) (x^{2} e^{x}) - 18 x^{3} (\sin (3 x^{2} e^{x}) (e^{2 x})) + 6 (x^{1 + 1} (- 6 \sin (3 x^{2} e^{x}) (e^{2 x}))) + 6 (\cos (3 x^{2} e^{x}) e^{x}) + 6 (\cos (3 x^{2} e^{x}) (x e^{x}))$

Etapa 2.4.9.23

Some $1$ e $1$ .

$- 9 x^{4} (\sin (3 x^{2} e^{x}) (e^{2 x})) - 18 x^{3} (\sin (3 x^{2} e^{x}) (e^{2 x})) + 6 \cos (3 x^{2} e^{x}) (e^{x} (x)) + 3 \cos (3 x^{2} e^{x}) (x^{2} e^{x}) - 18 x^{3} (\sin (3 x^{2} e^{x}) (e^{2 x})) + 6 (x^{2} (- 6 \sin (3 x^{2} e^{x}) (e^{2 x}))) + 6 (\cos (3 x^{2} e^{x}) e^{x}) + 6 (\cos (3 x^{2} e^{x}) (x e^{x}))$

Etapa 2.4.9.24

Multiplique $- 6$ por $6$ .

$- 9 x^{4} (\sin (3 x^{2} e^{x}) (e^{2 x})) - 18 x^{3} (\sin (3 x^{2} e^{x}) (e^{2 x})) + 6 \cos (3 x^{2} e^{x}) (e^{x} (x)) + 3 \cos (3 x^{2} e^{x}) (x^{2} e^{x}) - 18 x^{3} (\sin (3 x^{2} e^{x}) (e^{2 x})) - 36 (x^{2} (\sin (3 x^{2} e^{x}) (e^{2 x}))) + 6 (\cos (3 x^{2} e^{x}) e^{x}) + 6 (\cos (3 x^{2} e^{x}) (x e^{x}))$

Etapa 2.4.9.25

Subtraia $18 x^{3} (\sin (3 x^{2} e^{x}) (e^{2 x}))$ de $- 18 x^{3} (\sin (3 x^{2} e^{x}) (e^{2 x}))$ .

$- 9 x^{4} (\sin (3 x^{2} e^{x}) (e^{2 x})) - 36 x^{3} \sin (3 x^{2} e^{x}) e^{2 x} + 6 \cos (3 x^{2} e^{x}) (e^{x} (x)) + 3 \cos (3 x^{2} e^{x}) (x^{2} e^{x}) - 36 x^{2} (\sin (3 x^{2} e^{x}) (e^{2 x})) + 6 \cos (3 x^{2} e^{x}) e^{x} + 6 \cos (3 x^{2} e^{x}) (x e^{x})$

Etapa 2.4.9.26

Mova $\cos (3 x^{2} e^{x})$ .

$- 9 x^{4} (\sin (3 x^{2} e^{x}) (e^{2 x})) - 36 x^{3} \sin (3 x^{2} e^{x}) e^{2 x} + 6 x e^{x} \cos (3 x^{2} e^{x}) + 6 x e^{x} \cos (3 x^{2} e^{x}) + 3 \cos (3 x^{2} e^{x}) (x^{2} e^{x}) - 36 x^{2} (\sin (3 x^{2} e^{x}) (e^{2 x})) + 6 \cos (3 x^{2} e^{x}) e^{x}$

Etapa 2.4.9.27

Some $6 x e^{x} \cos (3 x^{2} e^{x})$ e $6 x e^{x} \cos (3 x^{2} e^{x})$ .

$- 9 x^{4} (\sin (3 x^{2} e^{x}) (e^{2 x})) - 36 x^{3} \sin (3 x^{2} e^{x}) e^{2 x} + 12 x e^{x} \cos (3 x^{2} e^{x}) + 3 \cos (3 x^{2} e^{x}) (x^{2} e^{x}) - 36 x^{2} (\sin (3 x^{2} e^{x}) (e^{2 x})) + 6 \cos (3 x^{2} e^{x}) e^{x}$

Etapa 2.4.10

Reordene os termos.

$- 9 e^{2 x} x^{4} \sin (3 x^{2} e^{x}) - 36 e^{2 x} x^{3} \sin (3 x^{2} e^{x}) + 12 e^{x} x \cos (3 x^{2} e^{x}) + 3 e^{x} x^{2} \cos (3 x^{2} e^{x}) - 36 e^{2 x} x^{2} \sin (3 x^{2} e^{x}) + 6 e^{x} \cos (3 x^{2} e^{x})$

Etapa 2.4.11

Reordene os fatores em $- 9 e^{2 x} x^{4} \sin (3 x^{2} e^{x}) - 36 e^{2 x} x^{3} \sin (3 x^{2} e^{x}) + 12 e^{x} x \cos (3 x^{2} e^{x}) + 3 e^{x} x^{2} \cos (3 x^{2} e^{x}) - 36 e^{2 x} x^{2} \sin (3 x^{2} e^{x}) + 6 e^{x} \cos (3 x^{2} e^{x})$ .

$f'' (x) = - 9 x^{4} e^{2 x} \sin (3 x^{2} e^{x}) - 36 x^{3} e^{2 x} \sin (3 x^{2} e^{x}) + 12 x e^{x} \cos (3 x^{2} e^{x}) + 3 x^{2} e^{x} \cos (3 x^{2} e^{x}) - 36 x^{2} e^{2 x} \sin (3 x^{2} e^{x}) + 6 e^{x} \cos (3 x^{2} e^{x})$