Cálculo Exemplos

$f (x) = x^{\frac{9}{2}} e^{x}$

Etapa 1

Encontre a primeira derivada.

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Etapa 1.1

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 $f (x) = x^{\frac{9}{2}}$ e $g (x) = e^{x}$ .

$f' (x) = x^{\frac{9}{2}} \frac{d}{d x} (e^{x}) + e^{x} \frac{d}{d x} (x^{\frac{9}{2}})$

Etapa 1.2

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

$f' (x) = x^{\frac{9}{2}} e^{x} + e^{x} \frac{d}{d x} (x^{\frac{9}{2}})$

Etapa 1.3

Diferencie usando a regra da multiplicação de potências, que determina que $\frac{d}{d x} [x^{n}]$ é $n x^{n - 1}$ , em que $n = \frac{9}{2}$ .

$f' (x) = x^{\frac{9}{2}} e^{x} + e^{x} (\frac{9}{2} x^{\frac{9}{2} - 1})$

Etapa 1.4

Para escrever $- 1$ como fração com um denominador comum, multiplique por $\frac{2}{2}$ .

$f' (x) = x^{\frac{9}{2}} e^{x} + e^{x} (\frac{9}{2} x^{\frac{9}{2} - 1 \cdot \frac{2}{2}})$

Etapa 1.5

Combine $- 1$ e $\frac{2}{2}$ .

$f' (x) = x^{\frac{9}{2}} e^{x} + e^{x} (\frac{9}{2} x^{\frac{9}{2} + \frac{- 1 \cdot 2}{2}})$

Etapa 1.6

Combine os numeradores em relação ao denominador comum.

$f' (x) = x^{\frac{9}{2}} e^{x} + e^{x} (\frac{9}{2} x^{\frac{9 - 1 \cdot 2}{2}})$

Etapa 1.7

Simplifique o numerador.

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Etapa 1.7.1

Multiplique $- 1$ por $2$ .

$f' (x) = x^{\frac{9}{2}} e^{x} + e^{x} (\frac{9}{2} x^{\frac{9 - 2}{2}})$

Etapa 1.7.2

Subtraia $2$ de $9$ .

$f' (x) = x^{\frac{9}{2}} e^{x} + e^{x} (\frac{9}{2} x^{\frac{7}{2}})$

Etapa 1.8

Combine $\frac{9}{2}$ e $x^{\frac{7}{2}}$ .

$f' (x) = x^{\frac{9}{2}} e^{x} + e^{x} (\frac{9 x^{\frac{7}{2}}}{2})$

Etapa 1.9

Combine $e^{x}$ e $\frac{9 x^{\frac{7}{2}}}{2}$ .

$f' (x) = x^{\frac{9}{2}} e^{x} + \frac{e^{x} (9 x^{\frac{7}{2}})}{2}$

Etapa 1.10

Simplifique.

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Etapa 1.10.1

Reordene os termos.

$f' (x) = e^{x} x^{\frac{9}{2}} + \frac{e^{x} (9 x^{\frac{7}{2}})}{2}$

Etapa 1.10.2

Mova $9$ para a esquerda de $e^{x}$ .

$f' (x) = e^{x} x^{\frac{9}{2}} + \frac{9 e^{x} x^{\frac{7}{2}}}{2}$

Etapa 2

Encontre a segunda derivada.

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Etapa 2.1

De acordo com a regra da soma, a derivada de $e^{x} x^{\frac{9}{2}} + \frac{9 e^{x} x^{\frac{7}{2}}}{2}$ com relação a $x$ é $\frac{d}{d x} [e^{x} x^{\frac{9}{2}}] + \frac{d}{d x} [\frac{9 e^{x} x^{\frac{7}{2}}}{2}]$ .

$f'' (x) = \frac{d}{d x} (e^{x} x^{\frac{9}{2}}) + \frac{d}{d x} (\frac{9 e^{x} x^{\frac{7}{2}}}{2})$

Etapa 2.2

Avalie $\frac{d}{d x} [e^{x} x^{\frac{9}{2}}]$ .

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Etapa 2.2.1

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 $f (x) = e^{x}$ e $g (x) = x^{\frac{9}{2}}$ .

$f'' (x) = e^{x} \frac{d}{d x} (x^{\frac{9}{2}}) + x^{\frac{9}{2}} \frac{d}{d x} (e^{x}) + \frac{d}{d x} (\frac{9 e^{x} x^{\frac{7}{2}}}{2})$

Etapa 2.2.2

Diferencie usando a regra da multiplicação de potências, que determina que $\frac{d}{d x} [x^{n}]$ é $n x^{n - 1}$ , em que $n = \frac{9}{2}$ .

$f'' (x) = e^{x} (\frac{9}{2} x^{\frac{9}{2} - 1}) + x^{\frac{9}{2}} \frac{d}{d x} (e^{x}) + \frac{d}{d x} (\frac{9 e^{x} x^{\frac{7}{2}}}{2})$

Etapa 2.2.3

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

$f'' (x) = e^{x} (\frac{9}{2} x^{\frac{9}{2} - 1}) + x^{\frac{9}{2}} e^{x} + \frac{d}{d x} (\frac{9 e^{x} x^{\frac{7}{2}}}{2})$

Etapa 2.2.4

Para escrever $- 1$ como fração com um denominador comum, multiplique por $\frac{2}{2}$ .

$f'' (x) = e^{x} (\frac{9}{2} x^{\frac{9}{2} - 1 \cdot \frac{2}{2}}) + x^{\frac{9}{2}} e^{x} + \frac{d}{d x} (\frac{9 e^{x} x^{\frac{7}{2}}}{2})$

Etapa 2.2.5

Combine $- 1$ e $\frac{2}{2}$ .

$f'' (x) = e^{x} (\frac{9}{2} x^{\frac{9}{2} + \frac{- 1 \cdot 2}{2}}) + x^{\frac{9}{2}} e^{x} + \frac{d}{d x} (\frac{9 e^{x} x^{\frac{7}{2}}}{2})$

Etapa 2.2.6

Combine os numeradores em relação ao denominador comum.

$f'' (x) = e^{x} (\frac{9}{2} x^{\frac{9 - 1 \cdot 2}{2}}) + x^{\frac{9}{2}} e^{x} + \frac{d}{d x} (\frac{9 e^{x} x^{\frac{7}{2}}}{2})$

Etapa 2.2.7

Simplifique o numerador.

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Etapa 2.2.7.1

Multiplique $- 1$ por $2$ .

$f'' (x) = e^{x} (\frac{9}{2} x^{\frac{9 - 2}{2}}) + x^{\frac{9}{2}} e^{x} + \frac{d}{d x} (\frac{9 e^{x} x^{\frac{7}{2}}}{2})$

Etapa 2.2.7.2

Subtraia $2$ de $9$ .

$f'' (x) = e^{x} (\frac{9}{2} x^{\frac{7}{2}}) + x^{\frac{9}{2}} e^{x} + \frac{d}{d x} (\frac{9 e^{x} x^{\frac{7}{2}}}{2})$

Etapa 2.2.8

Combine $\frac{9}{2}$ e $x^{\frac{7}{2}}$ .

$f'' (x) = e^{x} (\frac{9 x^{\frac{7}{2}}}{2}) + x^{\frac{9}{2}} e^{x} + \frac{d}{d x} (\frac{9 e^{x} x^{\frac{7}{2}}}{2})$

Etapa 2.2.9

Combine $e^{x}$ e $\frac{9 x^{\frac{7}{2}}}{2}$ .

$f'' (x) = \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{d}{d x} (\frac{9 e^{x} x^{\frac{7}{2}}}{2})$

Etapa 2.3

Avalie $\frac{d}{d x} [\frac{9 e^{x} x^{\frac{7}{2}}}{2}]$ .

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Etapa 2.3.1

Como $\frac{9}{2}$ é constante em relação a $x$ , a derivada de $\frac{9 e^{x} x^{\frac{7}{2}}}{2}$ em relação a $x$ é $\frac{9}{2} \frac{d}{d x} [e^{x} x^{\frac{7}{2}}]$ .

$f'' (x) = \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot \frac{d}{d x} (e^{x} x^{\frac{7}{2}})$

Etapa 2.3.2

$f'' (x) = \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (e^{x} \frac{d}{d x} (x^{\frac{7}{2}}) + x^{\frac{7}{2}} \frac{d}{d x} (e^{x}))$

Etapa 2.3.3

Diferencie usando a regra da multiplicação de potências, que determina que $\frac{d}{d x} [x^{n}]$ é $n x^{n - 1}$ , em que $n = \frac{7}{2}$ .

$f'' (x) = \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (e^{x} (\frac{7}{2} x^{\frac{7}{2} - 1}) + x^{\frac{7}{2}} \frac{d}{d x} (e^{x}))$

Etapa 2.3.4

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

$f'' (x) = \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (e^{x} (\frac{7}{2} x^{\frac{7}{2} - 1}) + x^{\frac{7}{2}} e^{x})$

Etapa 2.3.5

Para escrever $- 1$ como fração com um denominador comum, multiplique por $\frac{2}{2}$ .

$f'' (x) = \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (e^{x} (\frac{7}{2} x^{\frac{7}{2} - 1 \cdot \frac{2}{2}}) + x^{\frac{7}{2}} e^{x})$

Etapa 2.3.6

Combine $- 1$ e $\frac{2}{2}$ .

$f'' (x) = \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (e^{x} (\frac{7}{2} x^{\frac{7}{2} + \frac{- 1 \cdot 2}{2}}) + x^{\frac{7}{2}} e^{x})$

Etapa 2.3.7

Combine os numeradores em relação ao denominador comum.

$f'' (x) = \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (e^{x} (\frac{7}{2} x^{\frac{7 - 1 \cdot 2}{2}}) + x^{\frac{7}{2}} e^{x})$

Etapa 2.3.8

Simplifique o numerador.

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Etapa 2.3.8.1

Multiplique $- 1$ por $2$ .

$f'' (x) = \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (e^{x} (\frac{7}{2} x^{\frac{7 - 2}{2}}) + x^{\frac{7}{2}} e^{x})$

Etapa 2.3.8.2

Subtraia $2$ de $7$ .

$f'' (x) = \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (e^{x} (\frac{7}{2} x^{\frac{5}{2}}) + x^{\frac{7}{2}} e^{x})$

Etapa 2.3.9

Combine $\frac{7}{2}$ e $x^{\frac{5}{2}}$ .

$f'' (x) = \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (e^{x} (\frac{7 x^{\frac{5}{2}}}{2}) + x^{\frac{7}{2}} e^{x})$

Etapa 2.3.10

Combine $e^{x}$ e $\frac{7 x^{\frac{5}{2}}}{2}$ .

$f'' (x) = \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x})$

Etapa 2.4

Simplifique.

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Etapa 2.4.1

Aplique a propriedade distributiva.

$f'' (x) = \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot \frac{e^{x} (7 x^{\frac{5}{2}})}{2} + \frac{9}{2} \cdot (x^{\frac{7}{2}} e^{x})$

Etapa 2.4.2

Combine os termos.

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Etapa 2.4.2.1

Multiplique $\frac{9}{2}$ por $\frac{e^{x} (7 x^{\frac{5}{2}})}{2}$ .

$f'' (x) = \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9 (e^{x} (7 x^{\frac{5}{2}}))}{2 \cdot 2} + \frac{9}{2} \cdot (x^{\frac{7}{2}} e^{x})$

Etapa 2.4.2.2

Multiplique $7$ por $9$ .

$f'' (x) = \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{63 (e^{x} (x^{\frac{5}{2}}))}{2 \cdot 2} + \frac{9}{2} \cdot (x^{\frac{7}{2}} e^{x})$

Etapa 2.4.2.3

Multiplique $2$ por $2$ .

$f'' (x) = \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{63 (e^{x} (x^{\frac{5}{2}}))}{4} + \frac{9}{2} \cdot (x^{\frac{7}{2}} e^{x})$

Etapa 2.4.2.4

Combine $x^{\frac{7}{2}}$ e $\frac{9}{2}$ .

$f'' (x) = \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{63 e^{x} x^{\frac{5}{2}}}{4} + \frac{x^{\frac{7}{2}} \cdot 9}{2} \cdot e^{x}$

Etapa 2.4.2.5

Combine $\frac{x^{\frac{7}{2}} \cdot 9}{2}$ e $e^{x}$ .

$f'' (x) = \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{63 e^{x} x^{\frac{5}{2}}}{4} + \frac{x^{\frac{7}{2}} \cdot (9 e^{x})}{2}$

Etapa 2.4.2.6

Mova $9$ para a esquerda de $x^{\frac{7}{2}}$ .

$f'' (x) = \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{63 e^{x} x^{\frac{5}{2}}}{4} + \frac{9 \cdot (x^{\frac{7}{2}} e^{x})}{2}$

Etapa 2.4.2.7

Some $\frac{e^{x} (9 x^{\frac{7}{2}})}{2}$ e $\frac{9 x^{\frac{7}{2}} e^{x}}{2}$ .

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Etapa 2.4.2.7.1

Mova $e^{x}$ .

$f'' (x) = \frac{9 x^{\frac{7}{2}} e^{x}}{2} + \frac{9 x^{\frac{7}{2}} e^{x}}{2} + x^{\frac{9}{2}} e^{x} + \frac{63 e^{x} x^{\frac{5}{2}}}{4}$

Etapa 2.4.2.7.2

Some $\frac{9 x^{\frac{7}{2}} e^{x}}{2}$ e $\frac{9 x^{\frac{7}{2}} e^{x}}{2}$ .

$f'' (x) = 2 (\frac{9 x^{\frac{7}{2}} e^{x}}{2}) + x^{\frac{9}{2}} e^{x} + \frac{63 e^{x} x^{\frac{5}{2}}}{4}$

Etapa 2.4.2.8

Combine $2$ e $\frac{9 x^{\frac{7}{2}} e^{x}}{2}$ .

$f'' (x) = \frac{2 (9 x^{\frac{7}{2}} e^{x})}{2} + x^{\frac{9}{2}} e^{x} + \frac{63 e^{x} x^{\frac{5}{2}}}{4}$

Etapa 2.4.2.9

Multiplique $9$ por $2$ .

$f'' (x) = \frac{18 (x^{\frac{7}{2}} e^{x})}{2} + x^{\frac{9}{2}} e^{x} + \frac{63 e^{x} x^{\frac{5}{2}}}{4}$

Etapa 2.4.2.10

Fatore $2$ de $18 x^{\frac{7}{2}} e^{x}$ .

$f'' (x) = \frac{2 (9 x^{\frac{7}{2}} e^{x})}{2} + x^{\frac{9}{2}} e^{x} + \frac{63 e^{x} x^{\frac{5}{2}}}{4}$

Etapa 2.4.2.11

Cancele os fatores comuns.

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Etapa 2.4.2.11.1

Fatore $2$ de $2$ .

$f'' (x) = \frac{2 (9 x^{\frac{7}{2}} e^{x})}{2 (1)} + x^{\frac{9}{2}} e^{x} + \frac{63 e^{x} x^{\frac{5}{2}}}{4}$

Etapa 2.4.2.11.2

Cancele o fator comum.

$f'' (x) = \frac{2 (9 x^{\frac{7}{2}} e^{x})}{2 \cdot 1} + x^{\frac{9}{2}} e^{x} + \frac{63 e^{x} x^{\frac{5}{2}}}{4}$

Etapa 2.4.2.11.3

Reescreva a expressão.

$f'' (x) = \frac{9 x^{\frac{7}{2}} e^{x}}{1} + x^{\frac{9}{2}} e^{x} + \frac{63 e^{x} x^{\frac{5}{2}}}{4}$

Etapa 2.4.2.11.4

Divida $9 x^{\frac{7}{2}} e^{x}$ por $1$ .

$f'' (x) = 9 x^{\frac{7}{2}} e^{x} + x^{\frac{9}{2}} e^{x} + \frac{63 e^{x} x^{\frac{5}{2}}}{4}$

Etapa 2.4.3

Reordene os termos.

$f'' (x) = 9 e^{x} x^{\frac{7}{2}} + e^{x} x^{\frac{9}{2}} + \frac{63 e^{x} x^{\frac{5}{2}}}{4}$

Etapa 3

Encontre a terceira derivada.

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Etapa 3.1

De acordo com a regra da soma, a derivada de $9 e^{x} x^{\frac{7}{2}} + e^{x} x^{\frac{9}{2}} + \frac{63 e^{x} x^{\frac{5}{2}}}{4}$ com relação a $x$ é $\frac{d}{d x} [9 e^{x} x^{\frac{7}{2}}] + \frac{d}{d x} [e^{x} x^{\frac{9}{2}}] + \frac{d}{d x} [\frac{63 e^{x} x^{\frac{5}{2}}}{4}]$ .

$f''' (x) = \frac{d}{d x} (9 e^{x} x^{\frac{7}{2}}) + \frac{d}{d x} (e^{x} x^{\frac{9}{2}}) + \frac{d}{d x} (\frac{63 e^{x} x^{\frac{5}{2}}}{4})$

Etapa 3.2

Avalie $\frac{d}{d x} [9 e^{x} x^{\frac{7}{2}}]$ .

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Etapa 3.2.1

Como $9$ é constante em relação a $x$ , a derivada de $9 e^{x} x^{\frac{7}{2}}$ em relação a $x$ é $9 \frac{d}{d x} [e^{x} x^{\frac{7}{2}}]$ .

$f''' (x) = 9 \frac{d}{d x} (e^{x} x^{\frac{7}{2}}) + \frac{d}{d x} (e^{x} x^{\frac{9}{2}}) + \frac{d}{d x} (\frac{63 e^{x} x^{\frac{5}{2}}}{4})$

Etapa 3.2.2

$f''' (x) = 9 (e^{x} \frac{d}{d x} (x^{\frac{7}{2}}) + x^{\frac{7}{2}} \frac{d}{d x} (e^{x})) + \frac{d}{d x} (e^{x} x^{\frac{9}{2}}) + \frac{d}{d x} (\frac{63 e^{x} x^{\frac{5}{2}}}{4})$

Etapa 3.2.3

Diferencie usando a regra da multiplicação de potências, que determina que $\frac{d}{d x} [x^{n}]$ é $n x^{n - 1}$ , em que $n = \frac{7}{2}$ .

$f''' (x) = 9 (e^{x} (\frac{7}{2} x^{\frac{7}{2} - 1}) + x^{\frac{7}{2}} \frac{d}{d x} (e^{x})) + \frac{d}{d x} (e^{x} x^{\frac{9}{2}}) + \frac{d}{d x} (\frac{63 e^{x} x^{\frac{5}{2}}}{4})$

Etapa 3.2.4

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

$f''' (x) = 9 (e^{x} (\frac{7}{2} x^{\frac{7}{2} - 1}) + x^{\frac{7}{2}} e^{x}) + \frac{d}{d x} (e^{x} x^{\frac{9}{2}}) + \frac{d}{d x} (\frac{63 e^{x} x^{\frac{5}{2}}}{4})$

Etapa 3.2.5

Para escrever $- 1$ como fração com um denominador comum, multiplique por $\frac{2}{2}$ .

$f''' (x) = 9 (e^{x} (\frac{7}{2} x^{\frac{7}{2} - 1 \cdot \frac{2}{2}}) + x^{\frac{7}{2}} e^{x}) + \frac{d}{d x} (e^{x} x^{\frac{9}{2}}) + \frac{d}{d x} (\frac{63 e^{x} x^{\frac{5}{2}}}{4})$

Etapa 3.2.6

Combine $- 1$ e $\frac{2}{2}$ .

$f''' (x) = 9 (e^{x} (\frac{7}{2} x^{\frac{7}{2} + \frac{- 1 \cdot 2}{2}}) + x^{\frac{7}{2}} e^{x}) + \frac{d}{d x} (e^{x} x^{\frac{9}{2}}) + \frac{d}{d x} (\frac{63 e^{x} x^{\frac{5}{2}}}{4})$

Etapa 3.2.7

Combine os numeradores em relação ao denominador comum.

$f''' (x) = 9 (e^{x} (\frac{7}{2} x^{\frac{7 - 1 \cdot 2}{2}}) + x^{\frac{7}{2}} e^{x}) + \frac{d}{d x} (e^{x} x^{\frac{9}{2}}) + \frac{d}{d x} (\frac{63 e^{x} x^{\frac{5}{2}}}{4})$

Etapa 3.2.8

Simplifique o numerador.

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Etapa 3.2.8.1

Multiplique $- 1$ por $2$ .

$f''' (x) = 9 (e^{x} (\frac{7}{2} x^{\frac{7 - 2}{2}}) + x^{\frac{7}{2}} e^{x}) + \frac{d}{d x} (e^{x} x^{\frac{9}{2}}) + \frac{d}{d x} (\frac{63 e^{x} x^{\frac{5}{2}}}{4})$

Etapa 3.2.8.2

Subtraia $2$ de $7$ .

$f''' (x) = 9 (e^{x} (\frac{7}{2} x^{\frac{5}{2}}) + x^{\frac{7}{2}} e^{x}) + \frac{d}{d x} (e^{x} x^{\frac{9}{2}}) + \frac{d}{d x} (\frac{63 e^{x} x^{\frac{5}{2}}}{4})$

Etapa 3.2.9

Combine $\frac{7}{2}$ e $x^{\frac{5}{2}}$ .

$f''' (x) = 9 (e^{x} (\frac{7 x^{\frac{5}{2}}}{2}) + x^{\frac{7}{2}} e^{x}) + \frac{d}{d x} (e^{x} x^{\frac{9}{2}}) + \frac{d}{d x} (\frac{63 e^{x} x^{\frac{5}{2}}}{4})$

Etapa 3.2.10

Combine $e^{x}$ e $\frac{7 x^{\frac{5}{2}}}{2}$ .

$f''' (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{d}{d x} (e^{x} x^{\frac{9}{2}}) + \frac{d}{d x} (\frac{63 e^{x} x^{\frac{5}{2}}}{4})$

Etapa 3.3

Avalie $\frac{d}{d x} [e^{x} x^{\frac{9}{2}}]$ .

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Etapa 3.3.1

$f''' (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + e^{x} \frac{d}{d x} (x^{\frac{9}{2}}) + x^{\frac{9}{2}} \frac{d}{d x} (e^{x}) + \frac{d}{d x} (\frac{63 e^{x} x^{\frac{5}{2}}}{4})$

Etapa 3.3.2

Diferencie usando a regra da multiplicação de potências, que determina que $\frac{d}{d x} [x^{n}]$ é $n x^{n - 1}$ , em que $n = \frac{9}{2}$ .

$f''' (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + e^{x} (\frac{9}{2} x^{\frac{9}{2} - 1}) + x^{\frac{9}{2}} \frac{d}{d x} (e^{x}) + \frac{d}{d x} (\frac{63 e^{x} x^{\frac{5}{2}}}{4})$

Etapa 3.3.3

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

$f''' (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + e^{x} (\frac{9}{2} x^{\frac{9}{2} - 1}) + x^{\frac{9}{2}} e^{x} + \frac{d}{d x} (\frac{63 e^{x} x^{\frac{5}{2}}}{4})$

Etapa 3.3.4

Para escrever $- 1$ como fração com um denominador comum, multiplique por $\frac{2}{2}$ .

$f''' (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + e^{x} (\frac{9}{2} x^{\frac{9}{2} - 1 \cdot \frac{2}{2}}) + x^{\frac{9}{2}} e^{x} + \frac{d}{d x} (\frac{63 e^{x} x^{\frac{5}{2}}}{4})$

Etapa 3.3.5

Combine $- 1$ e $\frac{2}{2}$ .

$f''' (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + e^{x} (\frac{9}{2} x^{\frac{9}{2} + \frac{- 1 \cdot 2}{2}}) + x^{\frac{9}{2}} e^{x} + \frac{d}{d x} (\frac{63 e^{x} x^{\frac{5}{2}}}{4})$

Etapa 3.3.6

Combine os numeradores em relação ao denominador comum.

$f''' (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + e^{x} (\frac{9}{2} x^{\frac{9 - 1 \cdot 2}{2}}) + x^{\frac{9}{2}} e^{x} + \frac{d}{d x} (\frac{63 e^{x} x^{\frac{5}{2}}}{4})$

Etapa 3.3.7

Simplifique o numerador.

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Etapa 3.3.7.1

Multiplique $- 1$ por $2$ .

$f''' (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + e^{x} (\frac{9}{2} x^{\frac{9 - 2}{2}}) + x^{\frac{9}{2}} e^{x} + \frac{d}{d x} (\frac{63 e^{x} x^{\frac{5}{2}}}{4})$

Etapa 3.3.7.2

Subtraia $2$ de $9$ .

$f''' (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + e^{x} (\frac{9}{2} x^{\frac{7}{2}}) + x^{\frac{9}{2}} e^{x} + \frac{d}{d x} (\frac{63 e^{x} x^{\frac{5}{2}}}{4})$

Etapa 3.3.8

Combine $\frac{9}{2}$ e $x^{\frac{7}{2}}$ .

$f''' (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + e^{x} (\frac{9 x^{\frac{7}{2}}}{2}) + x^{\frac{9}{2}} e^{x} + \frac{d}{d x} (\frac{63 e^{x} x^{\frac{5}{2}}}{4})$

Etapa 3.3.9

Combine $e^{x}$ e $\frac{9 x^{\frac{7}{2}}}{2}$ .

$f''' (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{d}{d x} (\frac{63 e^{x} x^{\frac{5}{2}}}{4})$

Etapa 3.4

Avalie $\frac{d}{d x} [\frac{63 e^{x} x^{\frac{5}{2}}}{4}]$ .

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Etapa 3.4.1

Como $\frac{63}{4}$ é constante em relação a $x$ , a derivada de $\frac{63 e^{x} x^{\frac{5}{2}}}{4}$ em relação a $x$ é $\frac{63}{4} \frac{d}{d x} [e^{x} x^{\frac{5}{2}}]$ .

$f''' (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{63}{4} \cdot \frac{d}{d x} (e^{x} x^{\frac{5}{2}})$

Etapa 3.4.2

$f''' (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{63}{4} \cdot (e^{x} \frac{d}{d x} (x^{\frac{5}{2}}) + x^{\frac{5}{2}} \frac{d}{d x} (e^{x}))$

Etapa 3.4.3

Diferencie usando a regra da multiplicação de potências, que determina que $\frac{d}{d x} [x^{n}]$ é $n x^{n - 1}$ , em que $n = \frac{5}{2}$ .

$f''' (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{63}{4} \cdot (e^{x} (\frac{5}{2} x^{\frac{5}{2} - 1}) + x^{\frac{5}{2}} \frac{d}{d x} (e^{x}))$

Etapa 3.4.4

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

Etapa 3.4.5

Para escrever $- 1$ como fração com um denominador comum, multiplique por $\frac{2}{2}$ .

Etapa 3.4.6

Combine $- 1$ e $\frac{2}{2}$ .

Etapa 3.4.7

Combine os numeradores em relação ao denominador comum.

Etapa 3.4.8

Simplifique o numerador.

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Etapa 3.4.8.1

Multiplique $- 1$ por $2$ .

Etapa 3.4.8.2

Subtraia $2$ de $5$ .

Etapa 3.4.9

Combine $\frac{5}{2}$ e $x^{\frac{3}{2}}$ .

Etapa 3.4.10

Combine $e^{x}$ e $\frac{5 x^{\frac{3}{2}}}{2}$ .

$f''' (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{63}{4} \cdot (\frac{e^{x} (5 x^{\frac{3}{2}})}{2} + x^{\frac{5}{2}} e^{x})$

Etapa 3.5

Simplifique.

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Etapa 3.5.1

Aplique a propriedade distributiva.

$f''' (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2}) + 9 (x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{63}{4} \cdot (\frac{e^{x} (5 x^{\frac{3}{2}})}{2} + x^{\frac{5}{2}} e^{x})$

Etapa 3.5.2

Aplique a propriedade distributiva.

$f''' (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2}) + 9 (x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{63}{4} \cdot \frac{e^{x} (5 x^{\frac{3}{2}})}{2} + \frac{63}{4} \cdot (x^{\frac{5}{2}} e^{x})$

Etapa 3.5.3

Combine os termos.

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Etapa 3.5.3.1

Combine $9$ e $\frac{e^{x} (7 x^{\frac{5}{2}})}{2}$ .

$f''' (x) = \frac{9 (e^{x} (7 x^{\frac{5}{2}}))}{2} + 9 (x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{63}{4} \cdot \frac{e^{x} (5 x^{\frac{3}{2}})}{2} + \frac{63}{4} \cdot (x^{\frac{5}{2}} e^{x})$

Etapa 3.5.3.2

Multiplique $7$ por $9$ .

$f''' (x) = \frac{63 (e^{x} (x^{\frac{5}{2}}))}{2} + 9 (x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{63}{4} \cdot \frac{e^{x} (5 x^{\frac{3}{2}})}{2} + \frac{63}{4} \cdot (x^{\frac{5}{2}} e^{x})$

Etapa 3.5.3.3

Multiplique $\frac{63}{4}$ por $\frac{e^{x} (5 x^{\frac{3}{2}})}{2}$ .

$f''' (x) = \frac{63 e^{x} x^{\frac{5}{2}}}{2} + 9 x^{\frac{7}{2}} e^{x} + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{63 (e^{x} (5 x^{\frac{3}{2}}))}{4 \cdot 2} + \frac{63}{4} \cdot (x^{\frac{5}{2}} e^{x})$

Etapa 3.5.3.4

Multiplique $5$ por $63$ .

$f''' (x) = \frac{63 e^{x} x^{\frac{5}{2}}}{2} + 9 x^{\frac{7}{2}} e^{x} + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{315 (e^{x} (x^{\frac{3}{2}}))}{4 \cdot 2} + \frac{63}{4} \cdot (x^{\frac{5}{2}} e^{x})$

Etapa 3.5.3.5

Multiplique $4$ por $2$ .

Etapa 3.5.3.6

Combine $x^{\frac{5}{2}}$ e $\frac{63}{4}$ .

$f''' (x) = \frac{63 e^{x} x^{\frac{5}{2}}}{2} + 9 x^{\frac{7}{2}} e^{x} + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{315 e^{x} x^{\frac{3}{2}}}{8} + \frac{x^{\frac{5}{2}} \cdot 63}{4} \cdot e^{x}$

Etapa 3.5.3.7

Combine $\frac{x^{\frac{5}{2}} \cdot 63}{4}$ e $e^{x}$ .

Etapa 3.5.3.8

Mova $63$ para a esquerda de $x^{\frac{5}{2}}$ .

Etapa 3.5.3.9

Some $\frac{63 e^{x} x^{\frac{5}{2}}}{2}$ e $\frac{63 x^{\frac{5}{2}} e^{x}}{4}$ .

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Etapa 3.5.3.9.1

Mova $e^{x}$ .

$f''' (x) = 9 x^{\frac{7}{2}} e^{x} + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{315 e^{x} x^{\frac{3}{2}}}{8} + \frac{63 x^{\frac{5}{2}} e^{x}}{2} + \frac{63 x^{\frac{5}{2}} e^{x}}{4}$

Etapa 3.5.3.9.2

Para escrever $\frac{63 x^{\frac{5}{2}} e^{x}}{2}$ como fração com um denominador comum, multiplique por $\frac{2}{2}$ .

Etapa 3.5.3.9.3

Escreva cada expressão com um denominador comum de $4$ , multiplicando cada um por um fator apropriado de $1$ .

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Etapa 3.5.3.9.3.1

Multiplique $\frac{63 x^{\frac{5}{2}} e^{x}}{2}$ por $\frac{2}{2}$ .

$f''' (x) = 9 x^{\frac{7}{2}} e^{x} + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{315 e^{x} x^{\frac{3}{2}}}{8} + \frac{63 x^{\frac{5}{2}} e^{x} \cdot 2}{2 \cdot 2} + \frac{63 x^{\frac{5}{2}} e^{x}}{4}$

Etapa 3.5.3.9.3.2

Multiplique $2$ por $2$ .

Etapa 3.5.3.9.4

Combine os numeradores em relação ao denominador comum.

Etapa 3.5.3.10

Multiplique $2$ por $63$ .

$f''' (x) = 9 x^{\frac{7}{2}} e^{x} + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{315 e^{x} x^{\frac{3}{2}}}{8} + \frac{126 x^{\frac{5}{2}} e^{x} + 63 x^{\frac{5}{2}} e^{x}}{4}$

Etapa 3.5.3.11

Some $126 x^{\frac{5}{2}} e^{x}$ e $63 x^{\frac{5}{2}} e^{x}$ .

$f''' (x) = 9 x^{\frac{7}{2}} e^{x} + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{315 e^{x} x^{\frac{3}{2}}}{8} + \frac{189 x^{\frac{5}{2}} e^{x}}{4}$

Etapa 3.5.4

Reordene os termos.

$f''' (x) = 9 e^{x} x^{\frac{7}{2}} + e^{x} x^{\frac{9}{2}} + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + \frac{315 e^{x} x^{\frac{3}{2}}}{8} + \frac{189 x^{\frac{5}{2}} e^{x}}{4}$

Etapa 3.5.5

Simplifique cada termo.

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Etapa 3.5.5.1

Simplifique o numerador.

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Etapa 3.5.5.1.1

Reescreva.

$f''' (x) = 9 e^{x} x^{\frac{7}{2}} + e^{x} x^{\frac{9}{2}} + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + \frac{315 e^{x} x^{\frac{3}{2}}}{8} + \frac{189 x^{\frac{5}{2}} e^{x}}{4}$

Etapa 3.5.5.1.2

Remova os parênteses desnecessários.

$f''' (x) = 9 e^{x} x^{\frac{7}{2}} + e^{x} x^{\frac{9}{2}} + \frac{e^{x} \cdot (9 x^{\frac{7}{2}})}{2} + \frac{315 e^{x} x^{\frac{3}{2}}}{8} + \frac{189 x^{\frac{5}{2}} e^{x}}{4}$

Etapa 3.5.5.2

Mova $9$ para a esquerda de $e^{x}$ .

$f''' (x) = 9 e^{x} x^{\frac{7}{2}} + e^{x} x^{\frac{9}{2}} + \frac{9 e^{x} x^{\frac{7}{2}}}{2} + \frac{315 e^{x} x^{\frac{3}{2}}}{8} + \frac{189 x^{\frac{5}{2}} e^{x}}{4}$

Etapa 4

Encontre a quarta derivada.

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Etapa 4.1

De acordo com a regra da soma, a derivada de $9 e^{x} x^{\frac{7}{2}} + e^{x} x^{\frac{9}{2}} + \frac{9 e^{x} x^{\frac{7}{2}}}{2} + \frac{315 e^{x} x^{\frac{3}{2}}}{8} + \frac{189 x^{\frac{5}{2}} e^{x}}{4}$ com relação a $x$ é $\frac{d}{d x} [9 e^{x} x^{\frac{7}{2}}] + \frac{d}{d x} [e^{x} x^{\frac{9}{2}}] + \frac{d}{d x} [\frac{9 e^{x} x^{\frac{7}{2}}}{2}] + \frac{d}{d x} [\frac{315 e^{x} x^{\frac{3}{2}}}{8}] + \frac{d}{d x} [\frac{189 x^{\frac{5}{2}} e^{x}}{4}]$ .

$f^{4} (x) = \frac{d}{d x} (9 e^{x} x^{\frac{7}{2}}) + \frac{d}{d x} (e^{x} x^{\frac{9}{2}}) + \frac{d}{d x} (\frac{9 e^{x} x^{\frac{7}{2}}}{2}) + \frac{d}{d x} (\frac{315 e^{x} x^{\frac{3}{2}}}{8}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.2

Avalie $\frac{d}{d x} [9 e^{x} x^{\frac{7}{2}}]$ .

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Etapa 4.2.1

Como $9$ é constante em relação a $x$ , a derivada de $9 e^{x} x^{\frac{7}{2}}$ em relação a $x$ é $9 \frac{d}{d x} [e^{x} x^{\frac{7}{2}}]$ .

$f^{4} (x) = 9 \frac{d}{d x} (e^{x} x^{\frac{7}{2}}) + \frac{d}{d x} (e^{x} x^{\frac{9}{2}}) + \frac{d}{d x} (\frac{9 e^{x} x^{\frac{7}{2}}}{2}) + \frac{d}{d x} (\frac{315 e^{x} x^{\frac{3}{2}}}{8}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.2.2

$f^{4} (x) = 9 (e^{x} \frac{d}{d x} (x^{\frac{7}{2}}) + x^{\frac{7}{2}} \frac{d}{d x} (e^{x})) + \frac{d}{d x} (e^{x} x^{\frac{9}{2}}) + \frac{d}{d x} (\frac{9 e^{x} x^{\frac{7}{2}}}{2}) + \frac{d}{d x} (\frac{315 e^{x} x^{\frac{3}{2}}}{8}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.2.3

Diferencie usando a regra da multiplicação de potências, que determina que $\frac{d}{d x} [x^{n}]$ é $n x^{n - 1}$ , em que $n = \frac{7}{2}$ .

$f^{4} (x) = 9 (e^{x} (\frac{7}{2} x^{\frac{7}{2} - 1}) + x^{\frac{7}{2}} \frac{d}{d x} (e^{x})) + \frac{d}{d x} (e^{x} x^{\frac{9}{2}}) + \frac{d}{d x} (\frac{9 e^{x} x^{\frac{7}{2}}}{2}) + \frac{d}{d x} (\frac{315 e^{x} x^{\frac{3}{2}}}{8}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.2.4

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

$f^{4} (x) = 9 (e^{x} (\frac{7}{2} x^{\frac{7}{2} - 1}) + x^{\frac{7}{2}} e^{x}) + \frac{d}{d x} (e^{x} x^{\frac{9}{2}}) + \frac{d}{d x} (\frac{9 e^{x} x^{\frac{7}{2}}}{2}) + \frac{d}{d x} (\frac{315 e^{x} x^{\frac{3}{2}}}{8}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.2.5

Para escrever $- 1$ como fração com um denominador comum, multiplique por $\frac{2}{2}$ .

$f^{4} (x) = 9 (e^{x} (\frac{7}{2} x^{\frac{7}{2} - 1 \cdot \frac{2}{2}}) + x^{\frac{7}{2}} e^{x}) + \frac{d}{d x} (e^{x} x^{\frac{9}{2}}) + \frac{d}{d x} (\frac{9 e^{x} x^{\frac{7}{2}}}{2}) + \frac{d}{d x} (\frac{315 e^{x} x^{\frac{3}{2}}}{8}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.2.6

Combine $- 1$ e $\frac{2}{2}$ .

$f^{4} (x) = 9 (e^{x} (\frac{7}{2} x^{\frac{7}{2} + \frac{- 1 \cdot 2}{2}}) + x^{\frac{7}{2}} e^{x}) + \frac{d}{d x} (e^{x} x^{\frac{9}{2}}) + \frac{d}{d x} (\frac{9 e^{x} x^{\frac{7}{2}}}{2}) + \frac{d}{d x} (\frac{315 e^{x} x^{\frac{3}{2}}}{8}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.2.7

Combine os numeradores em relação ao denominador comum.

$f^{4} (x) = 9 (e^{x} (\frac{7}{2} x^{\frac{7 - 1 \cdot 2}{2}}) + x^{\frac{7}{2}} e^{x}) + \frac{d}{d x} (e^{x} x^{\frac{9}{2}}) + \frac{d}{d x} (\frac{9 e^{x} x^{\frac{7}{2}}}{2}) + \frac{d}{d x} (\frac{315 e^{x} x^{\frac{3}{2}}}{8}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.2.8

Simplifique o numerador.

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Etapa 4.2.8.1

Multiplique $- 1$ por $2$ .

$f^{4} (x) = 9 (e^{x} (\frac{7}{2} x^{\frac{7 - 2}{2}}) + x^{\frac{7}{2}} e^{x}) + \frac{d}{d x} (e^{x} x^{\frac{9}{2}}) + \frac{d}{d x} (\frac{9 e^{x} x^{\frac{7}{2}}}{2}) + \frac{d}{d x} (\frac{315 e^{x} x^{\frac{3}{2}}}{8}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.2.8.2

Subtraia $2$ de $7$ .

$f^{4} (x) = 9 (e^{x} (\frac{7}{2} x^{\frac{5}{2}}) + x^{\frac{7}{2}} e^{x}) + \frac{d}{d x} (e^{x} x^{\frac{9}{2}}) + \frac{d}{d x} (\frac{9 e^{x} x^{\frac{7}{2}}}{2}) + \frac{d}{d x} (\frac{315 e^{x} x^{\frac{3}{2}}}{8}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.2.9

Combine $\frac{7}{2}$ e $x^{\frac{5}{2}}$ .

$f^{4} (x) = 9 (e^{x} (\frac{7 x^{\frac{5}{2}}}{2}) + x^{\frac{7}{2}} e^{x}) + \frac{d}{d x} (e^{x} x^{\frac{9}{2}}) + \frac{d}{d x} (\frac{9 e^{x} x^{\frac{7}{2}}}{2}) + \frac{d}{d x} (\frac{315 e^{x} x^{\frac{3}{2}}}{8}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.2.10

Combine $e^{x}$ e $\frac{7 x^{\frac{5}{2}}}{2}$ .

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{d}{d x} (e^{x} x^{\frac{9}{2}}) + \frac{d}{d x} (\frac{9 e^{x} x^{\frac{7}{2}}}{2}) + \frac{d}{d x} (\frac{315 e^{x} x^{\frac{3}{2}}}{8}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.3

Avalie $\frac{d}{d x} [e^{x} x^{\frac{9}{2}}]$ .

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Etapa 4.3.1

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + e^{x} \frac{d}{d x} (x^{\frac{9}{2}}) + x^{\frac{9}{2}} \frac{d}{d x} (e^{x}) + \frac{d}{d x} (\frac{9 e^{x} x^{\frac{7}{2}}}{2}) + \frac{d}{d x} (\frac{315 e^{x} x^{\frac{3}{2}}}{8}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.3.2

Diferencie usando a regra da multiplicação de potências, que determina que $\frac{d}{d x} [x^{n}]$ é $n x^{n - 1}$ , em que $n = \frac{9}{2}$ .

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + e^{x} (\frac{9}{2} x^{\frac{9}{2} - 1}) + x^{\frac{9}{2}} \frac{d}{d x} (e^{x}) + \frac{d}{d x} (\frac{9 e^{x} x^{\frac{7}{2}}}{2}) + \frac{d}{d x} (\frac{315 e^{x} x^{\frac{3}{2}}}{8}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.3.3

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

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + e^{x} (\frac{9}{2} x^{\frac{9}{2} - 1}) + x^{\frac{9}{2}} e^{x} + \frac{d}{d x} (\frac{9 e^{x} x^{\frac{7}{2}}}{2}) + \frac{d}{d x} (\frac{315 e^{x} x^{\frac{3}{2}}}{8}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.3.4

Para escrever $- 1$ como fração com um denominador comum, multiplique por $\frac{2}{2}$ .

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + e^{x} (\frac{9}{2} x^{\frac{9}{2} - 1 \cdot \frac{2}{2}}) + x^{\frac{9}{2}} e^{x} + \frac{d}{d x} (\frac{9 e^{x} x^{\frac{7}{2}}}{2}) + \frac{d}{d x} (\frac{315 e^{x} x^{\frac{3}{2}}}{8}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.3.5

Combine $- 1$ e $\frac{2}{2}$ .

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + e^{x} (\frac{9}{2} x^{\frac{9}{2} + \frac{- 1 \cdot 2}{2}}) + x^{\frac{9}{2}} e^{x} + \frac{d}{d x} (\frac{9 e^{x} x^{\frac{7}{2}}}{2}) + \frac{d}{d x} (\frac{315 e^{x} x^{\frac{3}{2}}}{8}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.3.6

Combine os numeradores em relação ao denominador comum.

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + e^{x} (\frac{9}{2} x^{\frac{9 - 1 \cdot 2}{2}}) + x^{\frac{9}{2}} e^{x} + \frac{d}{d x} (\frac{9 e^{x} x^{\frac{7}{2}}}{2}) + \frac{d}{d x} (\frac{315 e^{x} x^{\frac{3}{2}}}{8}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.3.7

Simplifique o numerador.

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Etapa 4.3.7.1

Multiplique $- 1$ por $2$ .

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + e^{x} (\frac{9}{2} x^{\frac{9 - 2}{2}}) + x^{\frac{9}{2}} e^{x} + \frac{d}{d x} (\frac{9 e^{x} x^{\frac{7}{2}}}{2}) + \frac{d}{d x} (\frac{315 e^{x} x^{\frac{3}{2}}}{8}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.3.7.2

Subtraia $2$ de $9$ .

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + e^{x} (\frac{9}{2} x^{\frac{7}{2}}) + x^{\frac{9}{2}} e^{x} + \frac{d}{d x} (\frac{9 e^{x} x^{\frac{7}{2}}}{2}) + \frac{d}{d x} (\frac{315 e^{x} x^{\frac{3}{2}}}{8}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.3.8

Combine $\frac{9}{2}$ e $x^{\frac{7}{2}}$ .

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + e^{x} (\frac{9 x^{\frac{7}{2}}}{2}) + x^{\frac{9}{2}} e^{x} + \frac{d}{d x} (\frac{9 e^{x} x^{\frac{7}{2}}}{2}) + \frac{d}{d x} (\frac{315 e^{x} x^{\frac{3}{2}}}{8}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.3.9

Combine $e^{x}$ e $\frac{9 x^{\frac{7}{2}}}{2}$ .

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{d}{d x} (\frac{9 e^{x} x^{\frac{7}{2}}}{2}) + \frac{d}{d x} (\frac{315 e^{x} x^{\frac{3}{2}}}{8}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.4

Avalie $\frac{d}{d x} [\frac{9 e^{x} x^{\frac{7}{2}}}{2}]$ .

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Etapa 4.4.1

Como $\frac{9}{2}$ é constante em relação a $x$ , a derivada de $\frac{9 e^{x} x^{\frac{7}{2}}}{2}$ em relação a $x$ é $\frac{9}{2} \frac{d}{d x} [e^{x} x^{\frac{7}{2}}]$ .

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot \frac{d}{d x} (e^{x} x^{\frac{7}{2}}) + \frac{d}{d x} (\frac{315 e^{x} x^{\frac{3}{2}}}{8}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.4.2

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (e^{x} \frac{d}{d x} (x^{\frac{7}{2}}) + x^{\frac{7}{2}} \frac{d}{d x} (e^{x})) + \frac{d}{d x} (\frac{315 e^{x} x^{\frac{3}{2}}}{8}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.4.3

Diferencie usando a regra da multiplicação de potências, que determina que $\frac{d}{d x} [x^{n}]$ é $n x^{n - 1}$ , em que $n = \frac{7}{2}$ .

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (e^{x} (\frac{7}{2} x^{\frac{7}{2} - 1}) + x^{\frac{7}{2}} \frac{d}{d x} (e^{x})) + \frac{d}{d x} (\frac{315 e^{x} x^{\frac{3}{2}}}{8}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.4.4

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

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (e^{x} (\frac{7}{2} x^{\frac{7}{2} - 1}) + x^{\frac{7}{2}} e^{x}) + \frac{d}{d x} (\frac{315 e^{x} x^{\frac{3}{2}}}{8}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.4.5

Para escrever $- 1$ como fração com um denominador comum, multiplique por $\frac{2}{2}$ .

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (e^{x} (\frac{7}{2} x^{\frac{7}{2} - 1 \cdot \frac{2}{2}}) + x^{\frac{7}{2}} e^{x}) + \frac{d}{d x} (\frac{315 e^{x} x^{\frac{3}{2}}}{8}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.4.6

Combine $- 1$ e $\frac{2}{2}$ .

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (e^{x} (\frac{7}{2} x^{\frac{7}{2} + \frac{- 1 \cdot 2}{2}}) + x^{\frac{7}{2}} e^{x}) + \frac{d}{d x} (\frac{315 e^{x} x^{\frac{3}{2}}}{8}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.4.7

Combine os numeradores em relação ao denominador comum.

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (e^{x} (\frac{7}{2} x^{\frac{7 - 1 \cdot 2}{2}}) + x^{\frac{7}{2}} e^{x}) + \frac{d}{d x} (\frac{315 e^{x} x^{\frac{3}{2}}}{8}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.4.8

Simplifique o numerador.

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Etapa 4.4.8.1

Multiplique $- 1$ por $2$ .

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (e^{x} (\frac{7}{2} x^{\frac{7 - 2}{2}}) + x^{\frac{7}{2}} e^{x}) + \frac{d}{d x} (\frac{315 e^{x} x^{\frac{3}{2}}}{8}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.4.8.2

Subtraia $2$ de $7$ .

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (e^{x} (\frac{7}{2} x^{\frac{5}{2}}) + x^{\frac{7}{2}} e^{x}) + \frac{d}{d x} (\frac{315 e^{x} x^{\frac{3}{2}}}{8}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.4.9

Combine $\frac{7}{2}$ e $x^{\frac{5}{2}}$ .

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (e^{x} (\frac{7 x^{\frac{5}{2}}}{2}) + x^{\frac{7}{2}} e^{x}) + \frac{d}{d x} (\frac{315 e^{x} x^{\frac{3}{2}}}{8}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.4.10

Combine $e^{x}$ e $\frac{7 x^{\frac{5}{2}}}{2}$ .

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{d}{d x} (\frac{315 e^{x} x^{\frac{3}{2}}}{8}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.5

Avalie $\frac{d}{d x} [\frac{315 e^{x} x^{\frac{3}{2}}}{8}]$ .

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Etapa 4.5.1

Como $\frac{315}{8}$ é constante em relação a $x$ , a derivada de $\frac{315 e^{x} x^{\frac{3}{2}}}{8}$ em relação a $x$ é $\frac{315}{8} \frac{d}{d x} [e^{x} x^{\frac{3}{2}}]$ .

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{315}{8} \cdot \frac{d}{d x} (e^{x} x^{\frac{3}{2}}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.5.2

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{315}{8} \cdot (e^{x} \frac{d}{d x} (x^{\frac{3}{2}}) + x^{\frac{3}{2}} \frac{d}{d x} (e^{x})) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.5.3

Diferencie usando a regra da multiplicação de potências, que determina que $\frac{d}{d x} [x^{n}]$ é $n x^{n - 1}$ , em que $n = \frac{3}{2}$ .

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{315}{8} \cdot (e^{x} (\frac{3}{2} x^{\frac{3}{2} - 1}) + x^{\frac{3}{2}} \frac{d}{d x} (e^{x})) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.5.4

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

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{315}{8} \cdot (e^{x} (\frac{3}{2} x^{\frac{3}{2} - 1}) + x^{\frac{3}{2}} e^{x}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.5.5

Para escrever $- 1$ como fração com um denominador comum, multiplique por $\frac{2}{2}$ .

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{315}{8} \cdot (e^{x} (\frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}}) + x^{\frac{3}{2}} e^{x}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.5.6

Combine $- 1$ e $\frac{2}{2}$ .

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{315}{8} \cdot (e^{x} (\frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}}) + x^{\frac{3}{2}} e^{x}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.5.7

Combine os numeradores em relação ao denominador comum.

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{315}{8} \cdot (e^{x} (\frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}}) + x^{\frac{3}{2}} e^{x}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.5.8

Simplifique o numerador.

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Etapa 4.5.8.1

Multiplique $- 1$ por $2$ .

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{315}{8} \cdot (e^{x} (\frac{3}{2} x^{\frac{3 - 2}{2}}) + x^{\frac{3}{2}} e^{x}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.5.8.2

Subtraia $2$ de $3$ .

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{315}{8} \cdot (e^{x} (\frac{3}{2} x^{\frac{1}{2}}) + x^{\frac{3}{2}} e^{x}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.5.9

Combine $\frac{3}{2}$ e $x^{\frac{1}{2}}$ .

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{315}{8} \cdot (e^{x} (\frac{3 x^{\frac{1}{2}}}{2}) + x^{\frac{3}{2}} e^{x}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.5.10

Combine $e^{x}$ e $\frac{3 x^{\frac{1}{2}}}{2}$ .

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{315}{8} \cdot (\frac{e^{x} (3 x^{\frac{1}{2}})}{2} + x^{\frac{3}{2}} e^{x}) + \frac{d}{d x} (\frac{189 x^{\frac{5}{2}} e^{x}}{4})$

Etapa 4.6

Avalie $\frac{d}{d x} [\frac{189 x^{\frac{5}{2}} e^{x}}{4}]$ .

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Etapa 4.6.1

Como $\frac{189}{4}$ é constante em relação a $x$ , a derivada de $\frac{189 x^{\frac{5}{2}} e^{x}}{4}$ em relação a $x$ é $\frac{189}{4} \frac{d}{d x} [x^{\frac{5}{2}} e^{x}]$ .

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{315}{8} \cdot (\frac{e^{x} (3 x^{\frac{1}{2}})}{2} + x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot \frac{d}{d x} (x^{\frac{5}{2}} e^{x})$

Etapa 4.6.2

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 $f (x) = x^{\frac{5}{2}}$ e $g (x) = e^{x}$ .

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{315}{8} \cdot (\frac{e^{x} (3 x^{\frac{1}{2}})}{2} + x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} \frac{d}{d x} (e^{x}) + e^{x} \frac{d}{d x} (x^{\frac{5}{2}}))$

Etapa 4.6.3

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

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{315}{8} \cdot (\frac{e^{x} (3 x^{\frac{1}{2}})}{2} + x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x} + e^{x} \frac{d}{d x} (x^{\frac{5}{2}}))$

Etapa 4.6.4

Diferencie usando a regra da multiplicação de potências, que determina que $\frac{d}{d x} [x^{n}]$ é $n x^{n - 1}$ , em que $n = \frac{5}{2}$ .

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{315}{8} \cdot (\frac{e^{x} (3 x^{\frac{1}{2}})}{2} + x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x} + e^{x} (\frac{5}{2} x^{\frac{5}{2} - 1}))$

Etapa 4.6.5

Para escrever $- 1$ como fração com um denominador comum, multiplique por $\frac{2}{2}$ .

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{315}{8} \cdot (\frac{e^{x} (3 x^{\frac{1}{2}})}{2} + x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x} + e^{x} (\frac{5}{2} x^{\frac{5}{2} - 1 \cdot \frac{2}{2}}))$

Etapa 4.6.6

Combine $- 1$ e $\frac{2}{2}$ .

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{315}{8} \cdot (\frac{e^{x} (3 x^{\frac{1}{2}})}{2} + x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x} + e^{x} (\frac{5}{2} x^{\frac{5}{2} + \frac{- 1 \cdot 2}{2}}))$

Etapa 4.6.7

Combine os numeradores em relação ao denominador comum.

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{315}{8} \cdot (\frac{e^{x} (3 x^{\frac{1}{2}})}{2} + x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x} + e^{x} (\frac{5}{2} x^{\frac{5 - 1 \cdot 2}{2}}))$

Etapa 4.6.8

Simplifique o numerador.

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Etapa 4.6.8.1

Multiplique $- 1$ por $2$ .

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{315}{8} \cdot (\frac{e^{x} (3 x^{\frac{1}{2}})}{2} + x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x} + e^{x} (\frac{5}{2} x^{\frac{5 - 2}{2}}))$

Etapa 4.6.8.2

Subtraia $2$ de $5$ .

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{315}{8} \cdot (\frac{e^{x} (3 x^{\frac{1}{2}})}{2} + x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x} + e^{x} (\frac{5}{2} x^{\frac{3}{2}}))$

Etapa 4.6.9

Combine $\frac{5}{2}$ e $x^{\frac{3}{2}}$ .

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{315}{8} \cdot (\frac{e^{x} (3 x^{\frac{1}{2}})}{2} + x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x} + e^{x} (\frac{5 x^{\frac{3}{2}}}{2}))$

Etapa 4.6.10

Combine $e^{x}$ e $\frac{5 x^{\frac{3}{2}}}{2}$ .

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{315}{8} \cdot (\frac{e^{x} (3 x^{\frac{1}{2}})}{2} + x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x} + \frac{e^{x} (5 x^{\frac{3}{2}})}{2})$

Etapa 4.7

Simplifique.

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Etapa 4.7.1

Aplique a propriedade distributiva.

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2}) + 9 (x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot (\frac{e^{x} (7 x^{\frac{5}{2}})}{2} + x^{\frac{7}{2}} e^{x}) + \frac{315}{8} \cdot (\frac{e^{x} (3 x^{\frac{1}{2}})}{2} + x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x} + \frac{e^{x} (5 x^{\frac{3}{2}})}{2})$

Etapa 4.7.2

Aplique a propriedade distributiva.

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2}) + 9 (x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot \frac{e^{x} (7 x^{\frac{5}{2}})}{2} + \frac{9}{2} \cdot (x^{\frac{7}{2}} e^{x}) + \frac{315}{8} \cdot (\frac{e^{x} (3 x^{\frac{1}{2}})}{2} + x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x} + \frac{e^{x} (5 x^{\frac{3}{2}})}{2})$

Etapa 4.7.3

Aplique a propriedade distributiva.

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2}) + 9 (x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot \frac{e^{x} (7 x^{\frac{5}{2}})}{2} + \frac{9}{2} \cdot (x^{\frac{7}{2}} e^{x}) + \frac{315}{8} \cdot \frac{e^{x} (3 x^{\frac{1}{2}})}{2} + \frac{315}{8} \cdot (x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x} + \frac{e^{x} (5 x^{\frac{3}{2}})}{2})$

Etapa 4.7.4

Aplique a propriedade distributiva.

$f^{4} (x) = 9 (\frac{e^{x} (7 x^{\frac{5}{2}})}{2}) + 9 (x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot \frac{e^{x} (7 x^{\frac{5}{2}})}{2} + \frac{9}{2} \cdot (x^{\frac{7}{2}} e^{x}) + \frac{315}{8} \cdot \frac{e^{x} (3 x^{\frac{1}{2}})}{2} + \frac{315}{8} \cdot (x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x}) + \frac{189}{4} \cdot \frac{e^{x} (5 x^{\frac{3}{2}})}{2}$

Etapa 4.7.5

Combine os termos.

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Etapa 4.7.5.1

Combine $9$ e $\frac{e^{x} (7 x^{\frac{5}{2}})}{2}$ .

$f^{4} (x) = \frac{9 (e^{x} (7 x^{\frac{5}{2}}))}{2} + 9 (x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot \frac{e^{x} (7 x^{\frac{5}{2}})}{2} + \frac{9}{2} \cdot (x^{\frac{7}{2}} e^{x}) + \frac{315}{8} \cdot \frac{e^{x} (3 x^{\frac{1}{2}})}{2} + \frac{315}{8} \cdot (x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x}) + \frac{189}{4} \cdot \frac{e^{x} (5 x^{\frac{3}{2}})}{2}$

Etapa 4.7.5.2

Multiplique $7$ por $9$ .

$f^{4} (x) = \frac{63 (e^{x} (x^{\frac{5}{2}}))}{2} + 9 (x^{\frac{7}{2}} e^{x}) + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9}{2} \cdot \frac{e^{x} (7 x^{\frac{5}{2}})}{2} + \frac{9}{2} \cdot (x^{\frac{7}{2}} e^{x}) + \frac{315}{8} \cdot \frac{e^{x} (3 x^{\frac{1}{2}})}{2} + \frac{315}{8} \cdot (x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x}) + \frac{189}{4} \cdot \frac{e^{x} (5 x^{\frac{3}{2}})}{2}$

Etapa 4.7.5.3

Multiplique $\frac{9}{2}$ por $\frac{e^{x} (7 x^{\frac{5}{2}})}{2}$ .

$f^{4} (x) = \frac{63 e^{x} x^{\frac{5}{2}}}{2} + 9 x^{\frac{7}{2}} e^{x} + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{9 (e^{x} (7 x^{\frac{5}{2}}))}{2 \cdot 2} + \frac{9}{2} \cdot (x^{\frac{7}{2}} e^{x}) + \frac{315}{8} \cdot \frac{e^{x} (3 x^{\frac{1}{2}})}{2} + \frac{315}{8} \cdot (x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x}) + \frac{189}{4} \cdot \frac{e^{x} (5 x^{\frac{3}{2}})}{2}$

Etapa 4.7.5.4

Multiplique $7$ por $9$ .

$f^{4} (x) = \frac{63 e^{x} x^{\frac{5}{2}}}{2} + 9 x^{\frac{7}{2}} e^{x} + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{63 (e^{x} (x^{\frac{5}{2}}))}{2 \cdot 2} + \frac{9}{2} \cdot (x^{\frac{7}{2}} e^{x}) + \frac{315}{8} \cdot \frac{e^{x} (3 x^{\frac{1}{2}})}{2} + \frac{315}{8} \cdot (x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x}) + \frac{189}{4} \cdot \frac{e^{x} (5 x^{\frac{3}{2}})}{2}$

Etapa 4.7.5.5

Multiplique $2$ por $2$ .

$f^{4} (x) = \frac{63 e^{x} x^{\frac{5}{2}}}{2} + 9 x^{\frac{7}{2}} e^{x} + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{63 (e^{x} (x^{\frac{5}{2}}))}{4} + \frac{9}{2} \cdot (x^{\frac{7}{2}} e^{x}) + \frac{315}{8} \cdot \frac{e^{x} (3 x^{\frac{1}{2}})}{2} + \frac{315}{8} \cdot (x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x}) + \frac{189}{4} \cdot \frac{e^{x} (5 x^{\frac{3}{2}})}{2}$

Etapa 4.7.5.6

Combine $x^{\frac{7}{2}}$ e $\frac{9}{2}$ .

$f^{4} (x) = \frac{63 e^{x} x^{\frac{5}{2}}}{2} + 9 x^{\frac{7}{2}} e^{x} + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{63 e^{x} x^{\frac{5}{2}}}{4} + \frac{x^{\frac{7}{2}} \cdot 9}{2} \cdot e^{x} + \frac{315}{8} \cdot \frac{e^{x} (3 x^{\frac{1}{2}})}{2} + \frac{315}{8} \cdot (x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x}) + \frac{189}{4} \cdot \frac{e^{x} (5 x^{\frac{3}{2}})}{2}$

Etapa 4.7.5.7

Combine $\frac{x^{\frac{7}{2}} \cdot 9}{2}$ e $e^{x}$ .

$f^{4} (x) = \frac{63 e^{x} x^{\frac{5}{2}}}{2} + 9 x^{\frac{7}{2}} e^{x} + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{63 e^{x} x^{\frac{5}{2}}}{4} + \frac{x^{\frac{7}{2}} \cdot (9 e^{x})}{2} + \frac{315}{8} \cdot \frac{e^{x} (3 x^{\frac{1}{2}})}{2} + \frac{315}{8} \cdot (x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x}) + \frac{189}{4} \cdot \frac{e^{x} (5 x^{\frac{3}{2}})}{2}$

Etapa 4.7.5.8

Mova $9$ para a esquerda de $x^{\frac{7}{2}}$ .

$f^{4} (x) = \frac{63 e^{x} x^{\frac{5}{2}}}{2} + 9 x^{\frac{7}{2}} e^{x} + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{63 e^{x} x^{\frac{5}{2}}}{4} + \frac{9 \cdot (x^{\frac{7}{2}} e^{x})}{2} + \frac{315}{8} \cdot \frac{e^{x} (3 x^{\frac{1}{2}})}{2} + \frac{315}{8} \cdot (x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x}) + \frac{189}{4} \cdot \frac{e^{x} (5 x^{\frac{3}{2}})}{2}$

Etapa 4.7.5.9

Para escrever $\frac{63 e^{x} x^{\frac{5}{2}}}{2}$ como fração com um denominador comum, multiplique por $\frac{2}{2}$ .

$f^{4} (x) = 9 x^{\frac{7}{2}} e^{x} + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{63 e^{x} x^{\frac{5}{2}}}{2} \cdot \frac{2}{2} + \frac{63 e^{x} x^{\frac{5}{2}}}{4} + \frac{9 x^{\frac{7}{2}} e^{x}}{2} + \frac{315}{8} \cdot \frac{e^{x} (3 x^{\frac{1}{2}})}{2} + \frac{315}{8} \cdot (x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x}) + \frac{189}{4} \cdot \frac{e^{x} (5 x^{\frac{3}{2}})}{2}$

Etapa 4.7.5.10

Escreva cada expressão com um denominador comum de $4$ , multiplicando cada um por um fator apropriado de $1$ .

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Etapa 4.7.5.10.1

Multiplique $\frac{63 e^{x} x^{\frac{5}{2}}}{2}$ por $\frac{2}{2}$ .

$f^{4} (x) = 9 x^{\frac{7}{2}} e^{x} + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{63 e^{x} x^{\frac{5}{2}} \cdot 2}{2 \cdot 2} + \frac{63 e^{x} x^{\frac{5}{2}}}{4} + \frac{9 x^{\frac{7}{2}} e^{x}}{2} + \frac{315}{8} \cdot \frac{e^{x} (3 x^{\frac{1}{2}})}{2} + \frac{315}{8} \cdot (x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x}) + \frac{189}{4} \cdot \frac{e^{x} (5 x^{\frac{3}{2}})}{2}$

Etapa 4.7.5.10.2

Multiplique $2$ por $2$ .

$f^{4} (x) = 9 x^{\frac{7}{2}} e^{x} + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{63 e^{x} x^{\frac{5}{2}} \cdot 2}{4} + \frac{63 e^{x} x^{\frac{5}{2}}}{4} + \frac{9 x^{\frac{7}{2}} e^{x}}{2} + \frac{315}{8} \cdot \frac{e^{x} (3 x^{\frac{1}{2}})}{2} + \frac{315}{8} \cdot (x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x}) + \frac{189}{4} \cdot \frac{e^{x} (5 x^{\frac{3}{2}})}{2}$

Etapa 4.7.5.11

Combine os numeradores em relação ao denominador comum.

$f^{4} (x) = 9 x^{\frac{7}{2}} e^{x} + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{63 e^{x} x^{\frac{5}{2}} \cdot 2 + 63 e^{x} x^{\frac{5}{2}}}{4} + \frac{9 x^{\frac{7}{2}} e^{x}}{2} + \frac{315}{8} \cdot \frac{e^{x} (3 x^{\frac{1}{2}})}{2} + \frac{315}{8} \cdot (x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x}) + \frac{189}{4} \cdot \frac{e^{x} (5 x^{\frac{3}{2}})}{2}$

Etapa 4.7.5.12

Multiplique $2$ por $63$ .

$f^{4} (x) = 9 x^{\frac{7}{2}} e^{x} + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{126 e^{x} x^{\frac{5}{2}} + 63 e^{x} x^{\frac{5}{2}}}{4} + \frac{9 x^{\frac{7}{2}} e^{x}}{2} + \frac{315}{8} \cdot \frac{e^{x} (3 x^{\frac{1}{2}})}{2} + \frac{315}{8} \cdot (x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x}) + \frac{189}{4} \cdot \frac{e^{x} (5 x^{\frac{3}{2}})}{2}$

Etapa 4.7.5.13

Some $126 e^{x} x^{\frac{5}{2}}$ e $63 e^{x} x^{\frac{5}{2}}$ .

$f^{4} (x) = 9 x^{\frac{7}{2}} e^{x} + \frac{e^{x} (9 x^{\frac{7}{2}})}{2} + x^{\frac{9}{2}} e^{x} + \frac{189 e^{x} x^{\frac{5}{2}}}{4} + \frac{9 x^{\frac{7}{2}} e^{x}}{2} + \frac{315}{8} \cdot \frac{e^{x} (3 x^{\frac{1}{2}})}{2} + \frac{315}{8} \cdot (x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x}) + \frac{189}{4} \cdot \frac{e^{x} (5 x^{\frac{3}{2}})}{2}$

Etapa 4.7.5.14

Some $\frac{e^{x} (9 x^{\frac{7}{2}})}{2}$ e $\frac{9 x^{\frac{7}{2}} e^{x}}{2}$ .

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Etapa 4.7.5.14.1

Mova $e^{x}$ .

$f^{4} (x) = 9 x^{\frac{7}{2}} e^{x} + \frac{9 x^{\frac{7}{2}} e^{x}}{2} + \frac{9 x^{\frac{7}{2}} e^{x}}{2} + x^{\frac{9}{2}} e^{x} + \frac{189 e^{x} x^{\frac{5}{2}}}{4} + \frac{315}{8} \cdot \frac{e^{x} (3 x^{\frac{1}{2}})}{2} + \frac{315}{8} \cdot (x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x}) + \frac{189}{4} \cdot \frac{e^{x} (5 x^{\frac{3}{2}})}{2}$

Etapa 4.7.5.14.2

Some $\frac{9 x^{\frac{7}{2}} e^{x}}{2}$ e $\frac{9 x^{\frac{7}{2}} e^{x}}{2}$ .

$f^{4} (x) = 9 x^{\frac{7}{2}} e^{x} + 2 (\frac{9 x^{\frac{7}{2}} e^{x}}{2}) + x^{\frac{9}{2}} e^{x} + \frac{189 e^{x} x^{\frac{5}{2}}}{4} + \frac{315}{8} \cdot \frac{e^{x} (3 x^{\frac{1}{2}})}{2} + \frac{315}{8} \cdot (x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x}) + \frac{189}{4} \cdot \frac{e^{x} (5 x^{\frac{3}{2}})}{2}$

Etapa 4.7.5.15

Combine $2$ e $\frac{9 x^{\frac{7}{2}} e^{x}}{2}$ .

$f^{4} (x) = 9 x^{\frac{7}{2}} e^{x} + \frac{2 (9 x^{\frac{7}{2}} e^{x})}{2} + x^{\frac{9}{2}} e^{x} + \frac{189 e^{x} x^{\frac{5}{2}}}{4} + \frac{315}{8} \cdot \frac{e^{x} (3 x^{\frac{1}{2}})}{2} + \frac{315}{8} \cdot (x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x}) + \frac{189}{4} \cdot \frac{e^{x} (5 x^{\frac{3}{2}})}{2}$

Etapa 4.7.5.16

Multiplique $9$ por $2$ .

$f^{4} (x) = 9 x^{\frac{7}{2}} e^{x} + \frac{18 (x^{\frac{7}{2}} e^{x})}{2} + x^{\frac{9}{2}} e^{x} + \frac{189 e^{x} x^{\frac{5}{2}}}{4} + \frac{315}{8} \cdot \frac{e^{x} (3 x^{\frac{1}{2}})}{2} + \frac{315}{8} \cdot (x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x}) + \frac{189}{4} \cdot \frac{e^{x} (5 x^{\frac{3}{2}})}{2}$

Etapa 4.7.5.17

Fatore $2$ de $18 x^{\frac{7}{2}} e^{x}$ .

Etapa 4.7.5.18

Cancele os fatores comuns.

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Etapa 4.7.5.18.1

Fatore $2$ de $2$ .

$f^{4} (x) = 9 x^{\frac{7}{2}} e^{x} + \frac{2 (9 x^{\frac{7}{2}} e^{x})}{2 (1)} + x^{\frac{9}{2}} e^{x} + \frac{189 e^{x} x^{\frac{5}{2}}}{4} + \frac{315}{8} \cdot \frac{e^{x} (3 x^{\frac{1}{2}})}{2} + \frac{315}{8} \cdot (x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x}) + \frac{189}{4} \cdot \frac{e^{x} (5 x^{\frac{3}{2}})}{2}$

Etapa 4.7.5.18.2

Cancele o fator comum.

$f^{4} (x) = 9 x^{\frac{7}{2}} e^{x} + \frac{2 (9 x^{\frac{7}{2}} e^{x})}{2 \cdot 1} + x^{\frac{9}{2}} e^{x} + \frac{189 e^{x} x^{\frac{5}{2}}}{4} + \frac{315}{8} \cdot \frac{e^{x} (3 x^{\frac{1}{2}})}{2} + \frac{315}{8} \cdot (x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x}) + \frac{189}{4} \cdot \frac{e^{x} (5 x^{\frac{3}{2}})}{2}$

Etapa 4.7.5.18.3

Reescreva a expressão.

$f^{4} (x) = 9 x^{\frac{7}{2}} e^{x} + \frac{9 x^{\frac{7}{2}} e^{x}}{1} + x^{\frac{9}{2}} e^{x} + \frac{189 e^{x} x^{\frac{5}{2}}}{4} + \frac{315}{8} \cdot \frac{e^{x} (3 x^{\frac{1}{2}})}{2} + \frac{315}{8} \cdot (x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x}) + \frac{189}{4} \cdot \frac{e^{x} (5 x^{\frac{3}{2}})}{2}$

Etapa 4.7.5.18.4

Divida $9 x^{\frac{7}{2}} e^{x}$ por $1$ .

$f^{4} (x) = 9 x^{\frac{7}{2}} e^{x} + 9 x^{\frac{7}{2}} e^{x} + x^{\frac{9}{2}} e^{x} + \frac{189 e^{x} x^{\frac{5}{2}}}{4} + \frac{315}{8} \cdot \frac{e^{x} (3 x^{\frac{1}{2}})}{2} + \frac{315}{8} \cdot (x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x}) + \frac{189}{4} \cdot \frac{e^{x} (5 x^{\frac{3}{2}})}{2}$

Etapa 4.7.5.19

Some $9 x^{\frac{7}{2}} e^{x}$ e $9 x^{\frac{7}{2}} e^{x}$ .

$f^{4} (x) = 18 x^{\frac{7}{2}} e^{x} + x^{\frac{9}{2}} e^{x} + \frac{189 e^{x} x^{\frac{5}{2}}}{4} + \frac{315}{8} \cdot \frac{e^{x} (3 x^{\frac{1}{2}})}{2} + \frac{315}{8} \cdot (x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x}) + \frac{189}{4} \cdot \frac{e^{x} (5 x^{\frac{3}{2}})}{2}$

Etapa 4.7.5.20

Multiplique $\frac{315}{8}$ por $\frac{e^{x} (3 x^{\frac{1}{2}})}{2}$ .

$f^{4} (x) = 18 x^{\frac{7}{2}} e^{x} + x^{\frac{9}{2}} e^{x} + \frac{189 e^{x} x^{\frac{5}{2}}}{4} + \frac{315 (e^{x} (3 x^{\frac{1}{2}}))}{8 \cdot 2} + \frac{315}{8} \cdot (x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x}) + \frac{189}{4} \cdot \frac{e^{x} (5 x^{\frac{3}{2}})}{2}$

Etapa 4.7.5.21

Multiplique $3$ por $315$ .

$f^{4} (x) = 18 x^{\frac{7}{2}} e^{x} + x^{\frac{9}{2}} e^{x} + \frac{189 e^{x} x^{\frac{5}{2}}}{4} + \frac{945 (e^{x} (x^{\frac{1}{2}}))}{8 \cdot 2} + \frac{315}{8} \cdot (x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x}) + \frac{189}{4} \cdot \frac{e^{x} (5 x^{\frac{3}{2}})}{2}$

Etapa 4.7.5.22

Multiplique $8$ por $2$ .

$f^{4} (x) = 18 x^{\frac{7}{2}} e^{x} + x^{\frac{9}{2}} e^{x} + \frac{189 e^{x} x^{\frac{5}{2}}}{4} + \frac{945 (e^{x} (x^{\frac{1}{2}}))}{16} + \frac{315}{8} \cdot (x^{\frac{3}{2}} e^{x}) + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x}) + \frac{189}{4} \cdot \frac{e^{x} (5 x^{\frac{3}{2}})}{2}$

Etapa 4.7.5.23

Combine $x^{\frac{3}{2}}$ e $\frac{315}{8}$ .

$f^{4} (x) = 18 x^{\frac{7}{2}} e^{x} + x^{\frac{9}{2}} e^{x} + \frac{189 e^{x} x^{\frac{5}{2}}}{4} + \frac{945 e^{x} x^{\frac{1}{2}}}{16} + \frac{x^{\frac{3}{2}} \cdot 315}{8} \cdot e^{x} + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x}) + \frac{189}{4} \cdot \frac{e^{x} (5 x^{\frac{3}{2}})}{2}$

Etapa 4.7.5.24

Combine $\frac{x^{\frac{3}{2}} \cdot 315}{8}$ e $e^{x}$ .

$f^{4} (x) = 18 x^{\frac{7}{2}} e^{x} + x^{\frac{9}{2}} e^{x} + \frac{189 e^{x} x^{\frac{5}{2}}}{4} + \frac{945 e^{x} x^{\frac{1}{2}}}{16} + \frac{x^{\frac{3}{2}} \cdot (315 e^{x})}{8} + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x}) + \frac{189}{4} \cdot \frac{e^{x} (5 x^{\frac{3}{2}})}{2}$

Etapa 4.7.5.25

Mova $315$ para a esquerda de $x^{\frac{3}{2}}$ .

$f^{4} (x) = 18 x^{\frac{7}{2}} e^{x} + x^{\frac{9}{2}} e^{x} + \frac{189 e^{x} x^{\frac{5}{2}}}{4} + \frac{945 e^{x} x^{\frac{1}{2}}}{16} + \frac{315 \cdot (x^{\frac{3}{2}} e^{x})}{8} + \frac{189}{4} \cdot (x^{\frac{5}{2}} e^{x}) + \frac{189}{4} \cdot \frac{e^{x} (5 x^{\frac{3}{2}})}{2}$

Etapa 4.7.5.26

Combine $x^{\frac{5}{2}}$ e $\frac{189}{4}$ .

$f^{4} (x) = 18 x^{\frac{7}{2}} e^{x} + x^{\frac{9}{2}} e^{x} + \frac{189 e^{x} x^{\frac{5}{2}}}{4} + \frac{945 e^{x} x^{\frac{1}{2}}}{16} + \frac{315 x^{\frac{3}{2}} e^{x}}{8} + \frac{x^{\frac{5}{2}} \cdot 189}{4} \cdot e^{x} + \frac{189}{4} \cdot \frac{e^{x} (5 x^{\frac{3}{2}})}{2}$

Etapa 4.7.5.27

Combine $\frac{x^{\frac{5}{2}} \cdot 189}{4}$ e $e^{x}$ .

$f^{4} (x) = 18 x^{\frac{7}{2}} e^{x} + x^{\frac{9}{2}} e^{x} + \frac{189 e^{x} x^{\frac{5}{2}}}{4} + \frac{945 e^{x} x^{\frac{1}{2}}}{16} + \frac{315 x^{\frac{3}{2}} e^{x}}{8} + \frac{x^{\frac{5}{2}} \cdot (189 e^{x})}{4} + \frac{189}{4} \cdot \frac{e^{x} (5 x^{\frac{3}{2}})}{2}$

Etapa 4.7.5.28

Mova $189$ para a esquerda de $x^{\frac{5}{2}}$ .

$f^{4} (x) = 18 x^{\frac{7}{2}} e^{x} + x^{\frac{9}{2}} e^{x} + \frac{189 e^{x} x^{\frac{5}{2}}}{4} + \frac{945 e^{x} x^{\frac{1}{2}}}{16} + \frac{315 x^{\frac{3}{2}} e^{x}}{8} + \frac{189 \cdot (x^{\frac{5}{2}} e^{x})}{4} + \frac{189}{4} \cdot \frac{e^{x} (5 x^{\frac{3}{2}})}{2}$

Etapa 4.7.5.29

Multiplique $\frac{189}{4}$ por $\frac{e^{x} (5 x^{\frac{3}{2}})}{2}$ .

$f^{4} (x) = 18 x^{\frac{7}{2}} e^{x} + x^{\frac{9}{2}} e^{x} + \frac{189 e^{x} x^{\frac{5}{2}}}{4} + \frac{945 e^{x} x^{\frac{1}{2}}}{16} + \frac{315 x^{\frac{3}{2}} e^{x}}{8} + \frac{189 x^{\frac{5}{2}} e^{x}}{4} + \frac{189 (e^{x} (5 x^{\frac{3}{2}}))}{4 \cdot 2}$

Etapa 4.7.5.30

Multiplique $5$ por $189$ .

$f^{4} (x) = 18 x^{\frac{7}{2}} e^{x} + x^{\frac{9}{2}} e^{x} + \frac{189 e^{x} x^{\frac{5}{2}}}{4} + \frac{945 e^{x} x^{\frac{1}{2}}}{16} + \frac{315 x^{\frac{3}{2}} e^{x}}{8} + \frac{189 x^{\frac{5}{2}} e^{x}}{4} + \frac{945 (e^{x} (x^{\frac{3}{2}}))}{4 \cdot 2}$

Etapa 4.7.5.31

Multiplique $4$ por $2$ .

$f^{4} (x) = 18 x^{\frac{7}{2}} e^{x} + x^{\frac{9}{2}} e^{x} + \frac{189 e^{x} x^{\frac{5}{2}}}{4} + \frac{945 e^{x} x^{\frac{1}{2}}}{16} + \frac{315 x^{\frac{3}{2}} e^{x}}{8} + \frac{189 x^{\frac{5}{2}} e^{x}}{4} + \frac{945 (e^{x} (x^{\frac{3}{2}}))}{8}$

Etapa 4.7.5.32

Some $\frac{189 e^{x} x^{\frac{5}{2}}}{4}$ e $\frac{189 x^{\frac{5}{2}} e^{x}}{4}$ .

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Etapa 4.7.5.32.1

Mova $e^{x}$ .

$f^{4} (x) = 18 x^{\frac{7}{2}} e^{x} + x^{\frac{9}{2}} e^{x} + \frac{189 x^{\frac{5}{2}} e^{x}}{4} + \frac{189 x^{\frac{5}{2}} e^{x}}{4} + \frac{945 e^{x} x^{\frac{1}{2}}}{16} + \frac{315 x^{\frac{3}{2}} e^{x}}{8} + \frac{945 e^{x} x^{\frac{3}{2}}}{8}$

Etapa 4.7.5.32.2

Some $\frac{189 x^{\frac{5}{2}} e^{x}}{4}$ e $\frac{189 x^{\frac{5}{2}} e^{x}}{4}$ .

$f^{4} (x) = 18 x^{\frac{7}{2}} e^{x} + x^{\frac{9}{2}} e^{x} + 2 (\frac{189 x^{\frac{5}{2}} e^{x}}{4}) + \frac{945 e^{x} x^{\frac{1}{2}}}{16} + \frac{315 x^{\frac{3}{2}} e^{x}}{8} + \frac{945 e^{x} x^{\frac{3}{2}}}{8}$

Etapa 4.7.5.33

Combine $2$ e $\frac{189 x^{\frac{5}{2}} e^{x}}{4}$ .

$f^{4} (x) = 18 x^{\frac{7}{2}} e^{x} + x^{\frac{9}{2}} e^{x} + \frac{2 (189 x^{\frac{5}{2}} e^{x})}{4} + \frac{945 e^{x} x^{\frac{1}{2}}}{16} + \frac{315 x^{\frac{3}{2}} e^{x}}{8} + \frac{945 e^{x} x^{\frac{3}{2}}}{8}$

Etapa 4.7.5.34

Multiplique $189$ por $2$ .

$f^{4} (x) = 18 x^{\frac{7}{2}} e^{x} + x^{\frac{9}{2}} e^{x} + \frac{378 (x^{\frac{5}{2}} e^{x})}{4} + \frac{945 e^{x} x^{\frac{1}{2}}}{16} + \frac{315 x^{\frac{3}{2}} e^{x}}{8} + \frac{945 e^{x} x^{\frac{3}{2}}}{8}$

Etapa 4.7.5.35

Fatore $2$ de $378 x^{\frac{5}{2}} e^{x}$ .

Etapa 4.7.5.36

Cancele os fatores comuns.

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Etapa 4.7.5.36.1

Fatore $2$ de $4$ .

$f^{4} (x) = 18 x^{\frac{7}{2}} e^{x} + x^{\frac{9}{2}} e^{x} + \frac{2 (189 x^{\frac{5}{2}} e^{x})}{2 (2)} + \frac{945 e^{x} x^{\frac{1}{2}}}{16} + \frac{315 x^{\frac{3}{2}} e^{x}}{8} + \frac{945 e^{x} x^{\frac{3}{2}}}{8}$

Etapa 4.7.5.36.2

Cancele o fator comum.

$f^{4} (x) = 18 x^{\frac{7}{2}} e^{x} + x^{\frac{9}{2}} e^{x} + \frac{2 (189 x^{\frac{5}{2}} e^{x})}{2 \cdot 2} + \frac{945 e^{x} x^{\frac{1}{2}}}{16} + \frac{315 x^{\frac{3}{2}} e^{x}}{8} + \frac{945 e^{x} x^{\frac{3}{2}}}{8}$

Etapa 4.7.5.36.3

Reescreva a expressão.

$f^{4} (x) = 18 x^{\frac{7}{2}} e^{x} + x^{\frac{9}{2}} e^{x} + \frac{189 x^{\frac{5}{2}} e^{x}}{2} + \frac{945 e^{x} x^{\frac{1}{2}}}{16} + \frac{315 x^{\frac{3}{2}} e^{x}}{8} + \frac{945 e^{x} x^{\frac{3}{2}}}{8}$

Etapa 4.7.5.37

Some $\frac{315 x^{\frac{3}{2}} e^{x}}{8}$ e $\frac{945 e^{x} x^{\frac{3}{2}}}{8}$ .

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Etapa 4.7.5.37.1

Mova $e^{x}$ .

Etapa 4.7.5.37.2

Combine os numeradores em relação ao denominador comum.

Etapa 4.7.5.38

Some $315 x^{\frac{3}{2}} e^{x}$ e $945 x^{\frac{3}{2}} e^{x}$ .

$f^{4} (x) = 18 x^{\frac{7}{2}} e^{x} + x^{\frac{9}{2}} e^{x} + \frac{189 x^{\frac{5}{2}} e^{x}}{2} + \frac{945 e^{x} x^{\frac{1}{2}}}{16} + \frac{1260 x^{\frac{3}{2}} e^{x}}{8}$

Etapa 4.7.5.39

Fatore $4$ de $1260 x^{\frac{3}{2}} e^{x}$ .

$f^{4} (x) = 18 x^{\frac{7}{2}} e^{x} + x^{\frac{9}{2}} e^{x} + \frac{189 x^{\frac{5}{2}} e^{x}}{2} + \frac{945 e^{x} x^{\frac{1}{2}}}{16} + \frac{4 (315 x^{\frac{3}{2}} e^{x})}{8}$

Etapa 4.7.5.40

Cancele os fatores comuns.

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Etapa 4.7.5.40.1

Fatore $4$ de $8$ .

$f^{4} (x) = 18 x^{\frac{7}{2}} e^{x} + x^{\frac{9}{2}} e^{x} + \frac{189 x^{\frac{5}{2}} e^{x}}{2} + \frac{945 e^{x} x^{\frac{1}{2}}}{16} + \frac{4 (315 x^{\frac{3}{2}} e^{x})}{4 (2)}$

Etapa 4.7.5.40.2

Cancele o fator comum.

$f^{4} (x) = 18 x^{\frac{7}{2}} e^{x} + x^{\frac{9}{2}} e^{x} + \frac{189 x^{\frac{5}{2}} e^{x}}{2} + \frac{945 e^{x} x^{\frac{1}{2}}}{16} + \frac{4 (315 x^{\frac{3}{2}} e^{x})}{4 \cdot 2}$

Etapa 4.7.5.40.3

Reescreva a expressão.

$f^{4} (x) = 18 x^{\frac{7}{2}} e^{x} + x^{\frac{9}{2}} e^{x} + \frac{189 x^{\frac{5}{2}} e^{x}}{2} + \frac{945 e^{x} x^{\frac{1}{2}}}{16} + \frac{315 x^{\frac{3}{2}} e^{x}}{2}$

Etapa 4.7.6

Reordene os termos.

$f^{4} (x) = 18 e^{x} x^{\frac{7}{2}} + e^{x} x^{\frac{9}{2}} + \frac{189 x^{\frac{5}{2}} e^{x}}{2} + \frac{945 e^{x} x^{\frac{1}{2}}}{16} + \frac{315 x^{\frac{3}{2}} e^{x}}{2}$

Etapa 5

A quarta derivada de $f (x)$ com relação a $x$ é $18 e^{x} x^{\frac{7}{2}} + e^{x} x^{\frac{9}{2}} + \frac{189 x^{\frac{5}{2}} e^{x}}{2} + \frac{945 e^{x} x^{\frac{1}{2}}}{16} + \frac{315 x^{\frac{3}{2}} e^{x}}{2}$ .

$18 e^{x} x^{\frac{7}{2}} + e^{x} x^{\frac{9}{2}} + \frac{189 x^{\frac{5}{2}} e^{x}}{2} + \frac{945 e^{x} x^{\frac{1}{2}}}{16} + \frac{315 x^{\frac{3}{2}} e^{x}}{2}$