Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

Differentiate using the Product Rule which states that $\frac{d}{d n} [f (n) g (n)]$ is $f (n) \frac{d}{d n} [g (n)] + g (n) \frac{d}{d n} [f (n)]$ where $f (n) = n$ and $g (n) = (n + 1)!$ .

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

Step 1.2

Differentiate using the Power Rule.

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Step 1.2.1

Differentiate using the Power Rule which states that $\frac{d}{d n} [n^{n}]$ is $n \cdot n^{n - 1}$ where $n = 1$ .

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

Step 1.2.2

Multiply $(n + 1)!$ by $1$ .

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

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $n \frac{d}{d n} [(n + 1)!] + (n + 1)!$ with respect to $n$ is $\frac{d}{d n} [n \frac{d}{d n} [(n + 1)!]] + \frac{d}{d n} [(n + 1)!]$ .

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

Step 2.2

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

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Step 2.2.1

Differentiate using the Product Rule which states that $\frac{d}{d n} [f (n) g (n)]$ is $f (n) \frac{d}{d n} [g (n)] + g (n) \frac{d}{d n} [f (n)]$ where $f (n) = n$ and $g (n) = \frac{d}{d n} [(n + 1)!]$ .

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

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d n} [n^{n}]$ is $n \cdot n^{n - 1}$ where $n = 1$ .

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

Step 2.2.3

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

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

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $n \frac{d}{d n} [\frac{d}{d n} [(n + 1)!]] + \frac{d}{d n} [(n + 1)!] + \frac{d}{d n} [(n + 1)!]$ with respect to $n$ is $\frac{d}{d n} [n \frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]] + \frac{d}{d n} [\frac{d}{d n} [(n + 1)!]] + \frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]$ .

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

Step 3.2

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

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Step 3.2.1

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

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d n} [n^{n}]$ is $n \cdot n^{n - 1}$ where $n = 1$ .

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

Step 3.2.3

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

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

Step 4

Find the fourth derivative.

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Step 4.1

By the Sum Rule, the derivative of $n \frac{d}{d n} [\frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]] + \frac{d}{d n} [\frac{d}{d n} [(n + 1)!]] + \frac{d}{d n} [\frac{d}{d n} [(n + 1)!]] + \frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]$ with respect to $n$ is $\frac{d}{d n} [n \frac{d}{d n} [\frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]]] + \frac{d}{d n} [\frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]] + \frac{d}{d n} [\frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]] + \frac{d}{d n} [\frac{d}{d n} [\frac{d}{d n} [(n + 1)!]]]$ .

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

Step 4.2

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

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Step 4.2.1

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

Step 4.2.2

Differentiate using the Power Rule which states that $\frac{d}{d n} [n^{n}]$ is $n \cdot n^{n - 1}$ where $n = 1$ .

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

Step 4.2.3

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

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