Calculus Examples

$f (x) = {(\frac{v}{v^{3} + 1})}^{6}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d v} [f (g (v))]$ is $f' (g (v)) g' (v)$ where $f (v) = v^{6}$ and $g (v) = \frac{v}{v^{3} + 1}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{v}{v^{3} + 1}$ .

$\frac{d}{d u} [u^{6}] \frac{d}{d v} [\frac{v}{v^{3} + 1}]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 6$ .

$6 u^{5} \frac{d}{d v} [\frac{v}{v^{3} + 1}]$

Step 1.3

Replace all occurrences of $u$ with $\frac{v}{v^{3} + 1}$ .

$6 {(\frac{v}{v^{3} + 1})}^{5} \frac{d}{d v} [\frac{v}{v^{3} + 1}]$

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d v} [\frac{f (v)}{g (v)}]$ is $\frac{g (v) \frac{d}{d v} [f (v)] - f (v) \frac{d}{d v} [g (v)]}{{g (v)}^{2}}$ where $f (v) = v$ and $g (v) = v^{3} + 1$ .

$6 {(\frac{v}{v^{3} + 1})}^{5} \frac{(v^{3} + 1) \frac{d}{d v} [v] - v \frac{d}{d v} [v^{3} + 1]}{{(v^{3} + 1)}^{2}}$

Step 3

Differentiate.

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Step 3.1

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

$6 {(\frac{v}{v^{3} + 1})}^{5} \frac{(v^{3} + 1) \cdot 1 - v \frac{d}{d v} [v^{3} + 1]}{{(v^{3} + 1)}^{2}}$

Step 3.2

Multiply $v^{3} + 1$ by $1$ .

$6 {(\frac{v}{v^{3} + 1})}^{5} \frac{v^{3} + 1 - v \frac{d}{d v} [v^{3} + 1]}{{(v^{3} + 1)}^{2}}$

Step 3.3

By the Sum Rule, the derivative of $v^{3} + 1$ with respect to $v$ is $\frac{d}{d v} [v^{3}] + \frac{d}{d v} [1]$ .

$6 {(\frac{v}{v^{3} + 1})}^{5} \frac{v^{3} + 1 - v (\frac{d}{d v} [v^{3}] + \frac{d}{d v} [1])}{{(v^{3} + 1)}^{2}}$

Step 3.4

Differentiate using the Power Rule which states that $\frac{d}{d v} [v^{n}]$ is $n v^{n - 1}$ where $n = 3$ .

$6 {(\frac{v}{v^{3} + 1})}^{5} \frac{v^{3} + 1 - v (3 v^{2} + \frac{d}{d v} [1])}{{(v^{3} + 1)}^{2}}$

Step 3.5

Since $1$ is constant with respect to $v$ , the derivative of $1$ with respect to $v$ is $0$ .

$6 {(\frac{v}{v^{3} + 1})}^{5} \frac{v^{3} + 1 - v (3 v^{2} + 0)}{{(v^{3} + 1)}^{2}}$

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $3 v^{2}$ and $0$ .

$6 {(\frac{v}{v^{3} + 1})}^{5} \frac{v^{3} + 1 - v (3 v^{2})}{{(v^{3} + 1)}^{2}}$

Step 3.6.2

Multiply $3$ by $- 1$ .

$6 {(\frac{v}{v^{3} + 1})}^{5} \frac{v^{3} + 1 - 3 v \cdot v^{2}}{{(v^{3} + 1)}^{2}}$

Step 4

Multiply $v$ by $v^{2}$ by adding the exponents.

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Step 4.1

Move $v^{2}$ .

$6 {(\frac{v}{v^{3} + 1})}^{5} \frac{v^{3} + 1 - 3 (v^{2} v)}{{(v^{3} + 1)}^{2}}$

Step 4.2

Multiply $v^{2}$ by $v$ .

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Step 4.2.1

Raise $v$ to the power of $1$ .

$6 {(\frac{v}{v^{3} + 1})}^{5} \frac{v^{3} + 1 - 3 (v^{2} v^{1})}{{(v^{3} + 1)}^{2}}$

Step 4.2.2

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

$6 {(\frac{v}{v^{3} + 1})}^{5} \frac{v^{3} + 1 - 3 v^{2 + 1}}{{(v^{3} + 1)}^{2}}$

Step 4.3

Add $2$ and $1$ .

$6 {(\frac{v}{v^{3} + 1})}^{5} \frac{v^{3} + 1 - 3 v^{3}}{{(v^{3} + 1)}^{2}}$

Step 5

Subtract $3 v^{3}$ from $v^{3}$ .

$6 {(\frac{v}{v^{3} + 1})}^{5} \frac{- 2 v^{3} + 1}{{(v^{3} + 1)}^{2}}$

Step 6

Simplify.

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Step 6.1

Apply the product rule to $\frac{v}{v^{3} + 1}$ .

$6 \frac{v^{5}}{{(v^{3} + 1)}^{5}} \frac{- 2 v^{3} + 1}{{(v^{3} + 1)}^{2}}$

Step 6.2

Combine terms.

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Step 6.2.1

Combine $6$ and $\frac{v^{5}}{{(v^{3} + 1)}^{5}}$ .

$\frac{6 v^{5}}{{(v^{3} + 1)}^{5}} \cdot \frac{- 2 v^{3} + 1}{{(v^{3} + 1)}^{2}}$

Step 6.2.2

Multiply $\frac{6 v^{5}}{{(v^{3} + 1)}^{5}}$ by $\frac{- 2 v^{3} + 1}{{(v^{3} + 1)}^{2}}$ .

$\frac{6 v^{5} (- 2 v^{3} + 1)}{{(v^{3} + 1)}^{5} {(v^{3} + 1)}^{2}}$

Step 6.2.3

Multiply ${(v^{3} + 1)}^{5}$ by ${(v^{3} + 1)}^{2}$ by adding the exponents.

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Step 6.2.3.1

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

$\frac{6 v^{5} (- 2 v^{3} + 1)}{{(v^{3} + 1)}^{5 + 2}}$

Step 6.2.3.2

Add $5$ and $2$ .

$\frac{6 v^{5} (- 2 v^{3} + 1)}{{(v^{3} + 1)}^{7}}$