Calculus Examples

${(\frac{5 x}{6 - 5 x^{3}})}^{3}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = \frac{5 x}{6 - 5 x^{3}}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{5 x}{6 - 5 x^{3}}$ .

$\frac{d}{d u} [u^{3}] \frac{d}{d x} [\frac{5 x}{6 - 5 x^{3}}]$

Step 1.2

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

$3 u^{2} \frac{d}{d x} [\frac{5 x}{6 - 5 x^{3}}]$

Step 1.3

Replace all occurrences of $u$ with $\frac{5 x}{6 - 5 x^{3}}$ .

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

Step 2

Differentiate using the Constant Multiple Rule.

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Step 2.1

Since $5$ is constant with respect to $x$ , the derivative of $\frac{5 x}{6 - 5 x^{3}}$ with respect to $x$ is $5 \frac{d}{d x} [\frac{x}{6 - 5 x^{3}}]$ .

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

Step 2.2

Multiply $5$ by $3$ .

$15 {(\frac{5 x}{6 - 5 x^{3}})}^{2} \frac{d}{d x} [\frac{x}{6 - 5 x^{3}}]$

Step 3

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x$ and $g (x) = 6 - 5 x^{3}$ .

$15 {(\frac{5 x}{6 - 5 x^{3}})}^{2} \frac{(6 - 5 x^{3}) \frac{d}{d x} [x] - x \frac{d}{d x} [6 - 5 x^{3}]}{{(6 - 5 x^{3})}^{2}}$

Step 4

Differentiate.

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Step 4.1

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

$15 {(\frac{5 x}{6 - 5 x^{3}})}^{2} \frac{(6 - 5 x^{3}) \cdot 1 - x \frac{d}{d x} [6 - 5 x^{3}]}{{(6 - 5 x^{3})}^{2}}$

Step 4.2

Multiply $6 - 5 x^{3}$ by $1$ .

$15 {(\frac{5 x}{6 - 5 x^{3}})}^{2} \frac{6 - 5 x^{3} - x \frac{d}{d x} [6 - 5 x^{3}]}{{(6 - 5 x^{3})}^{2}}$

Step 4.3

By the Sum Rule, the derivative of $6 - 5 x^{3}$ with respect to $x$ is $\frac{d}{d x} [6] + \frac{d}{d x} [- 5 x^{3}]$ .

$15 {(\frac{5 x}{6 - 5 x^{3}})}^{2} \frac{6 - 5 x^{3} - x (\frac{d}{d x} [6] + \frac{d}{d x} [- 5 x^{3}])}{{(6 - 5 x^{3})}^{2}}$

Step 4.4

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$15 {(\frac{5 x}{6 - 5 x^{3}})}^{2} \frac{6 - 5 x^{3} - x (0 + \frac{d}{d x} [- 5 x^{3}])}{{(6 - 5 x^{3})}^{2}}$

Step 4.5

Add $0$ and $\frac{d}{d x} [- 5 x^{3}]$ .

$15 {(\frac{5 x}{6 - 5 x^{3}})}^{2} \frac{6 - 5 x^{3} - x \frac{d}{d x} [- 5 x^{3}]}{{(6 - 5 x^{3})}^{2}}$

Step 4.6

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x^{3}$ with respect to $x$ is $- 5 \frac{d}{d x} [x^{3}]$ .

$15 {(\frac{5 x}{6 - 5 x^{3}})}^{2} \frac{6 - 5 x^{3} - x (- 5 \frac{d}{d x} [x^{3}])}{{(6 - 5 x^{3})}^{2}}$

Step 4.7

Multiply $- 5$ by $- 1$ .

$15 {(\frac{5 x}{6 - 5 x^{3}})}^{2} \frac{6 - 5 x^{3} + 5 x \frac{d}{d x} [x^{3}]}{{(6 - 5 x^{3})}^{2}}$

Step 4.8

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

$15 {(\frac{5 x}{6 - 5 x^{3}})}^{2} \frac{6 - 5 x^{3} + 5 x (3 x^{2})}{{(6 - 5 x^{3})}^{2}}$

Step 4.9

Multiply $3$ by $5$ .

$15 {(\frac{5 x}{6 - 5 x^{3}})}^{2} \frac{6 - 5 x^{3} + 15 x \cdot x^{2}}{{(6 - 5 x^{3})}^{2}}$

Step 5

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

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Step 5.1

Move $x^{2}$ .

$15 {(\frac{5 x}{6 - 5 x^{3}})}^{2} \frac{6 - 5 x^{3} + 15 (x^{2} x)}{{(6 - 5 x^{3})}^{2}}$

Step 5.2

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

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Step 5.2.1

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

$15 {(\frac{5 x}{6 - 5 x^{3}})}^{2} \frac{6 - 5 x^{3} + 15 (x^{2} x^{1})}{{(6 - 5 x^{3})}^{2}}$

Step 5.2.2

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

$15 {(\frac{5 x}{6 - 5 x^{3}})}^{2} \frac{6 - 5 x^{3} + 15 x^{2 + 1}}{{(6 - 5 x^{3})}^{2}}$

Step 5.3

Add $2$ and $1$ .

$15 {(\frac{5 x}{6 - 5 x^{3}})}^{2} \frac{6 - 5 x^{3} + 15 x^{3}}{{(6 - 5 x^{3})}^{2}}$

Step 6

Add $- 5 x^{3}$ and $15 x^{3}$ .

$15 {(\frac{5 x}{6 - 5 x^{3}})}^{2} \frac{6 + 10 x^{3}}{{(6 - 5 x^{3})}^{2}}$

Step 7

Combine $\frac{6 + 10 x^{3}}{{(6 - 5 x^{3})}^{2}}$ and $15$ .

$\frac{(6 + 10 x^{3}) \cdot 15}{{(6 - 5 x^{3})}^{2}} {(\frac{5 x}{6 - 5 x^{3}})}^{2}$

Step 8

Move $15$ to the left of $6 + 10 x^{3}$ .

$\frac{15 \cdot (6 + 10 x^{3})}{{(6 - 5 x^{3})}^{2}} {(\frac{5 x}{6 - 5 x^{3}})}^{2}$

Step 9

Simplify.

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Step 9.1

Apply the product rule to $\frac{5 x}{6 - 5 x^{3}}$ .

$\frac{15 (6 + 10 x^{3})}{{(6 - 5 x^{3})}^{2}} \cdot \frac{{(5 x)}^{2}}{{(6 - 5 x^{3})}^{2}}$

Step 9.2

Apply the product rule to $5 x$ .

$\frac{15 (6 + 10 x^{3})}{{(6 - 5 x^{3})}^{2}} \cdot \frac{5^{2} x^{2}}{{(6 - 5 x^{3})}^{2}}$

Step 9.3

Apply the distributive property.

$\frac{15 \cdot 6 + 15 (10 x^{3})}{{(6 - 5 x^{3})}^{2}} \cdot \frac{5^{2} x^{2}}{{(6 - 5 x^{3})}^{2}}$

Step 9.4

Combine terms.

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Step 9.4.1

Multiply $15$ by $6$ .

$\frac{90 + 15 (10 x^{3})}{{(6 - 5 x^{3})}^{2}} \cdot \frac{5^{2} x^{2}}{{(6 - 5 x^{3})}^{2}}$

Step 9.4.2

Multiply $10$ by $15$ .

$\frac{90 + 150 x^{3}}{{(6 - 5 x^{3})}^{2}} \cdot \frac{5^{2} x^{2}}{{(6 - 5 x^{3})}^{2}}$

Step 9.4.3

Raise $5$ to the power of $2$ .

$\frac{90 + 150 x^{3}}{{(6 - 5 x^{3})}^{2}} \cdot \frac{25 x^{2}}{{(6 - 5 x^{3})}^{2}}$

Step 9.4.4

Multiply $\frac{90 + 150 x^{3}}{{(6 - 5 x^{3})}^{2}}$ by $\frac{25 x^{2}}{{(6 - 5 x^{3})}^{2}}$ .

$\frac{(90 + 150 x^{3}) (25 x^{2})}{{(6 - 5 x^{3})}^{2} {(6 - 5 x^{3})}^{2}}$

Step 9.4.5

Multiply ${(6 - 5 x^{3})}^{2}$ by ${(6 - 5 x^{3})}^{2}$ by adding the exponents.

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Step 9.4.5.1

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

$\frac{(90 + 150 x^{3}) (25 x^{2})}{{(6 - 5 x^{3})}^{2 + 2}}$

Step 9.4.5.2

Add $2$ and $2$ .

$\frac{(90 + 150 x^{3}) (25 x^{2})}{{(6 - 5 x^{3})}^{4}}$

Step 9.4.6

Move $25$ to the left of $90 + 150 x^{3}$ .

$\frac{25 (90 + 150 x^{3}) x^{2}}{{(6 - 5 x^{3})}^{4}}$

Step 9.5

Reorder terms.

$\frac{25 x^{2} (90 + 150 x^{3})}{{(- 5 x^{3} + 6)}^{4}}$

Step 9.6

Simplify the numerator.

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Step 9.6.1

Factor $30$ out of $90 + 150 x^{3}$ .

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Step 9.6.1.1

Factor $30$ out of $90$ .

$\frac{25 x^{2} (30 (3) + 150 x^{3})}{{(- 5 x^{3} + 6)}^{4}}$

Step 9.6.1.2

Factor $30$ out of $150 x^{3}$ .

$\frac{25 x^{2} (30 (3) + 30 (5 x^{3}))}{{(- 5 x^{3} + 6)}^{4}}$

Step 9.6.1.3

Factor $30$ out of $30 (3) + 30 (5 x^{3})$ .

$\frac{25 x^{2} (30 (3 + 5 x^{3}))}{{(- 5 x^{3} + 6)}^{4}}$

$\frac{25 x^{2} \cdot 30 (3 + 5 x^{3})}{{(- 5 x^{3} + 6)}^{4}}$

Step 9.6.2

Multiply $30$ by $25$ .

$\frac{750 x^{2} (3 + 5 x^{3})}{{(- 5 x^{3} + 6)}^{4}}$