Calculus Examples

${(\frac{n (n + 1)}{2 n^{4}})}^{2}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d n} [f (g (n))]$ is $f' (g (n)) g' (n)$ where $f (n) = n^{2}$ and $g (n) = \frac{n (n + 1)}{2 n^{4}}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{n (n + 1)}{2 n^{4}}$ .

$\frac{d}{d u} [u^{2}] \frac{d}{d n} [\frac{n (n + 1)}{2 n^{4}}]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

$2 u \frac{d}{d n} [\frac{n (n + 1)}{2 n^{4}}]$

Step 1.3

Replace all occurrences of $u$ with $\frac{n (n + 1)}{2 n^{4}}$ .

$2 \frac{n (n + 1)}{2 n^{4}} \frac{d}{d n} [\frac{n (n + 1)}{2 n^{4}}]$

Step 2

Differentiate using the Constant Multiple Rule.

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Step 2.1

Combine $2$ and $\frac{n (n + 1)}{2 n^{4}}$ .

$\frac{2 n (n + 1)}{2 n^{4}} \frac{d}{d n} [\frac{n (n + 1)}{2 n^{4}}]$

Step 2.2

Cancel the common factor of $2$ .

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Step 2.2.1

Cancel the common factor.

$\frac{2 n (n + 1)}{2 n^{4}} \frac{d}{d n} [\frac{n (n + 1)}{2 n^{4}}]$

Step 2.2.2

Rewrite the expression.

$\frac{n (n + 1)}{n^{4}} \frac{d}{d n} [\frac{n (n + 1)}{2 n^{4}}]$

Step 2.3

Since $\frac{1}{2}$ is constant with respect to $n$ , the derivative of $\frac{n (n + 1)}{2 n^{4}}$ with respect to $n$ is $\frac{1}{2} \frac{d}{d n} [\frac{n (n + 1)}{n^{4}}]$ .

$\frac{n (n + 1)}{n^{4}} (\frac{1}{2} \frac{d}{d n} [\frac{n (n + 1)}{n^{4}}])$

Step 2.4

Multiply $\frac{1}{2}$ by $\frac{n (n + 1)}{n^{4}}$ .

$\frac{n (n + 1)}{2 n^{4}} \frac{d}{d n} [\frac{n (n + 1)}{n^{4}}]$

Step 3

Differentiate using the Quotient Rule which states that $\frac{d}{d n} [\frac{f (n)}{g (n)}]$ is $\frac{g (n) \frac{d}{d n} [f (n)] - f (n) \frac{d}{d n} [g (n)]}{{g (n)}^{2}}$ where $f (n) = n (n + 1)$ and $g (n) = n^{4}$ .

$\frac{n (n + 1)}{2 n^{4}} \cdot \frac{n^{4} \frac{d}{d n} [n (n + 1)] - n (n + 1) \frac{d}{d n} [n^{4}]}{{(n^{4})}^{2}}$

Step 4

Differentiate using the Power Rule.

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Step 4.1

Multiply the exponents in ${(n^{4})}^{2}$ .

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Step 4.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{n (n + 1)}{2 n^{4}} \cdot \frac{n^{4} \frac{d}{d n} [n (n + 1)] - n (n + 1) \frac{d}{d n} [n^{4}]}{n^{4 \cdot 2}}$

Step 4.1.2

Multiply $4$ by $2$ .

$\frac{n (n + 1)}{2 n^{4}} \cdot \frac{n^{4} \frac{d}{d n} [n (n + 1)] - n (n + 1) \frac{d}{d n} [n^{4}]}{n^{8}}$

Step 4.2

Differentiate using the Power Rule which states that $\frac{d}{d n} [n^{n}]$ is $n \cdot n^{n - 1}$ where $n = 4$ .

$\frac{n (n + 1)}{2 n^{4}} \cdot \frac{n^{4} \frac{d}{d n} [n (n + 1)] - n (n + 1) (4 n^{3})}{n^{8}}$

Step 4.3

Simplify with factoring out.

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Step 4.3.1

Multiply $4$ by $- 1$ .

$\frac{n (n + 1)}{2 n^{4}} \cdot \frac{n^{4} \frac{d}{d n} [n (n + 1)] - 4 n (n + 1) n^{3}}{n^{8}}$

Step 4.3.2

Factor $n^{3}$ out of $n^{4} \frac{d}{d n} [n (n + 1)] - 4 n (n + 1) n^{3}$ .

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Step 4.3.2.1

Factor $n^{3}$ out of $n^{4} \frac{d}{d n} [n (n + 1)]$ .

$\frac{n (n + 1)}{2 n^{4}} \cdot \frac{n^{3} (n \frac{d}{d n} [n (n + 1)]) - 4 n (n + 1) n^{3}}{n^{8}}$

Step 4.3.2.2

Factor $n^{3}$ out of $- 4 n (n + 1) n^{3}$ .

$\frac{n (n + 1)}{2 n^{4}} \cdot \frac{n^{3} (n \frac{d}{d n} [n (n + 1)]) + n^{3} (- 4 n (n + 1))}{n^{8}}$

Step 4.3.2.3

Factor $n^{3}$ out of $n^{3} (n \frac{d}{d n} [n (n + 1)]) + n^{3} (- 4 n (n + 1))$ .

$\frac{n (n + 1)}{2 n^{4}} \cdot \frac{n^{3} (n \frac{d}{d n} [n (n + 1)] - 4 n (n + 1))}{n^{8}}$

Step 5

Cancel the common factors.

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Step 5.1

Factor $n^{3}$ out of $n^{8}$ .

$\frac{n (n + 1)}{2 n^{4}} \cdot \frac{n^{3} (n \frac{d}{d n} [n (n + 1)] - 4 n (n + 1))}{n^{3} n^{5}}$

Step 5.2

Cancel the common factor.

$\frac{n (n + 1)}{2 n^{4}} \cdot \frac{n^{3} (n \frac{d}{d n} [n (n + 1)] - 4 n (n + 1))}{n^{3} n^{5}}$

Step 5.3

Rewrite the expression.

$\frac{n (n + 1)}{2 n^{4}} \cdot \frac{n \frac{d}{d n} [n (n + 1)] - 4 n (n + 1)}{n^{5}}$

Step 6

Simplify.

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Step 6.1

Multiply $\frac{n (n + 1)}{2 n^{4}}$ by $\frac{n \frac{d}{d n} [n (n + 1)] - 4 n (n + 1)}{n^{5}}$ .

$\frac{n (n + 1) (n \frac{d}{d n} [n (n + 1)] - 4 n (n + 1))}{2 n^{4} n^{5}}$

Step 6.2

Multiply $n^{4}$ by $n^{5}$ by adding the exponents.

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Step 6.2.1

Move $n^{5}$ .

$\frac{n (n + 1) (n \frac{d}{d n} [n (n + 1)] - 4 n (n + 1))}{2 (n^{5} n^{4})}$

Step 6.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{n (n + 1) (n \frac{d}{d n} [n (n + 1)] - 4 n (n + 1))}{2 n^{5 + 4}}$

Step 6.2.3

Add $5$ and $4$ .

$\frac{n (n + 1) (n \frac{d}{d n} [n (n + 1)] - 4 n (n + 1))}{2 n^{9}}$