Cálculo Ejemplos

$a_{n} = n \cdot (n + 1)!$

Paso 1

Obtén la primera derivada.

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Paso 1.1

Diferencia con la regla del producto, que establece que $\frac{d}{d n} [f (n) g (n)]$ es $f (n) \frac{d}{d n} [g (n)] + g (n) \frac{d}{d n} [f (n)]$ donde $f (n) = n$ y $g (n) = (n + 1)!$ .

$n \frac{d}{d n} [(n + 1)!] + (n + 1)! \frac{d}{d n} [n]$

Paso 1.2

Diferencia con la regla de la potencia.

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Paso 1.2.1

Diferencia con la regla de la potencia, que establece que $\frac{d}{d n} [n^{n}]$ es $n \cdot n^{n - 1}$ donde $n = 1$ .

$n \frac{d}{d n} [(n + 1)!] + (n + 1)! \cdot 1$

Paso 1.2.2

Multiplica $(n + 1)!$ por $1$ .

$f' (n) = n \frac{d}{d n} [(n + 1)!] + (n + 1)!$

Paso 2

Obtener la segunda derivada.

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Paso 2.1

Según la regla de la suma, la derivada de $n \frac{d}{d n} [(n + 1)!] + (n + 1)!$ con respecto a $n$ es $\frac{d}{d n} [n \frac{d}{d n} [(n + 1)!]] + \frac{d}{d n} [(n + 1)!]$ .

$\frac{d}{d n} [n \frac{d}{d n} [(n + 1)!]] + \frac{d}{d n} [(n + 1)!]$

Paso 2.2

Evalúa $\frac{d}{d n} [n \frac{d}{d n} [(n + 1)!]]$ .

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Paso 2.2.1

Diferencia con la regla del producto, que establece que $\frac{d}{d n} [f (n) g (n)]$ es $f (n) \frac{d}{d n} [g (n)] + g (n) \frac{d}{d n} [f (n)]$ donde $f (n) = n$ y $g (n) = \frac{d}{d n} [(n + 1)!]$ .

$n \frac{d}{d n} [\frac{d}{d n} [(n + 1)!]] + \frac{d}{d n} [(n + 1)!] \frac{d}{d n} [n] + \frac{d}{d n} [(n + 1)!]$

Paso 2.2.2

Diferencia con la regla de la potencia, que establece que $\frac{d}{d n} [n^{n}]$ es $n \cdot n^{n - 1}$ donde $n = 1$ .

$n \frac{d}{d n} [\frac{d}{d n} [(n + 1)!]] + \frac{d}{d n} [(n + 1)!] \cdot 1 + \frac{d}{d n} [(n + 1)!]$

Paso 2.2.3

Multiplica $\frac{d}{d n} [(n + 1)!]$ por $1$ .

$f'' (n) = n \frac{d}{d n} [\frac{d}{d n} [(n + 1)!]] + \frac{d}{d n} [(n + 1)!] + \frac{d}{d n} [(n + 1)!]$

Paso 3

Obtén la tercera derivada.

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Paso 3.1

Según la regla de la suma, la derivada de $n \frac{d}{d n} [\frac{d}{d n} [(n + 1)!]] + \frac{d}{d n} [(n + 1)!] + \frac{d}{d n} [(n + 1)!]$ con respecto a $n$ es $\frac{d}{d n} [n \frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]] + \frac{d}{d n} [\frac{d}{d n} [(n + 1)!]] + \frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]$ .

$\frac{d}{d n} [n \frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]] + \frac{d}{d n} [\frac{d}{d n} [(n + 1)!]] + \frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]$

Paso 3.2

Evalúa $\frac{d}{d n} [n \frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]]$ .

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Paso 3.2.1

$n \frac{d}{d n} [\frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]] + \frac{d}{d n} [\frac{d}{d n} [(n + 1)!]] \frac{d}{d n} [n] + \frac{d}{d n} [\frac{d}{d n} [(n + 1)!]] + \frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]$

Paso 3.2.2

Diferencia con la regla de la potencia, que establece que $\frac{d}{d n} [n^{n}]$ es $n \cdot n^{n - 1}$ donde $n = 1$ .

$n \frac{d}{d n} [\frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]] + \frac{d}{d n} [\frac{d}{d n} [(n + 1)!]] \cdot 1 + \frac{d}{d n} [\frac{d}{d n} [(n + 1)!]] + \frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]$

Paso 3.2.3

Multiplica $\frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]$ por $1$ .

$f''' (n) = n \frac{d}{d n} [\frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]] + \frac{d}{d n} [\frac{d}{d n} [(n + 1)!]] + \frac{d}{d n} [\frac{d}{d n} [(n + 1)!]] + \frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]$

Paso 4

Obtén la cuarta derivada.

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Paso 4.1

Según la regla de la suma, la derivada de $n \frac{d}{d n} [\frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]] + \frac{d}{d n} [\frac{d}{d n} [(n + 1)!]] + \frac{d}{d n} [\frac{d}{d n} [(n + 1)!]] + \frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]$ con respecto a $n$ es $\frac{d}{d n} [n \frac{d}{d n} [\frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]]] + \frac{d}{d n} [\frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]] + \frac{d}{d n} [\frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]] + \frac{d}{d n} [\frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]]$ .

$\frac{d}{d n} [n \frac{d}{d n} [\frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]]] + \frac{d}{d n} [\frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]] + \frac{d}{d n} [\frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]] + \frac{d}{d n} [\frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]]$

Paso 4.2

Evalúa $\frac{d}{d n} [n \frac{d}{d n} [\frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]]]$ .

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Paso 4.2.1

$n \frac{d}{d n} [\frac{d}{d n} [\frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]]] + \frac{d}{d n} [\frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]] \frac{d}{d n} [n] + \frac{d}{d n} [\frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]] + \frac{d}{d n} [\frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]] + \frac{d}{d n} [\frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]]$

Paso 4.2.2

Diferencia con la regla de la potencia, que establece que $\frac{d}{d n} [n^{n}]$ es $n \cdot n^{n - 1}$ donde $n = 1$ .

$n \frac{d}{d n} [\frac{d}{d n} [\frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]]] + \frac{d}{d n} [\frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]] \cdot 1 + \frac{d}{d n} [\frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]] + \frac{d}{d n} [\frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]] + \frac{d}{d n} [\frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]]$

Paso 4.2.3

Multiplica $\frac{d}{d n} [\frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]]$ por $1$ .

$f^{4} (n) = n \frac{d}{d n} [\frac{d}{d n} [\frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]]] + \frac{d}{d n} [\frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]] + \frac{d}{d n} [\frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]] + \frac{d}{d n} [\frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]] + \frac{d}{d n} [\frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]]$