Algebra Examples

$\frac{27 x^{3} - 18 x}{9 x}$

Step 1

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 $f (x) = 27 x^{3} - 18 x$ and $g (x) = 9 x$ .

$\frac{9 x \frac{d}{d x} [27 x^{3} - 18 x] - (27 x^{3} - 18 x) \frac{d}{d x} [9 x]}{{(9 x)}^{2}}$

Step 2

By the Sum Rule, the derivative of $27 x^{3} - 18 x$ with respect to $x$ is $\frac{d}{d x} [27 x^{3}] + \frac{d}{d x} [- 18 x]$ .

$\frac{9 x (\frac{d}{d x} [27 x^{3}] + \frac{d}{d x} [- 18 x]) - (27 x^{3} - 18 x) \frac{d}{d x} [9 x]}{{(9 x)}^{2}}$

Step 3

Evaluate $\frac{d}{d x} [27 x^{3}]$ .

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Step 3.1

Since $27$ is constant with respect to $x$ , the derivative of $27 x^{3}$ with respect to $x$ is $27 \frac{d}{d x} [x^{3}]$ .

$\frac{9 x (27 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 18 x]) - (27 x^{3} - 18 x) \frac{d}{d x} [9 x]}{{(9 x)}^{2}}$

Step 3.2

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

$\frac{9 x (27 (3 x^{2}) + \frac{d}{d x} [- 18 x]) - (27 x^{3} - 18 x) \frac{d}{d x} [9 x]}{{(9 x)}^{2}}$

Step 3.3

Multiply $3$ by $27$ .

$\frac{9 x (81 x^{2} + \frac{d}{d x} [- 18 x]) - (27 x^{3} - 18 x) \frac{d}{d x} [9 x]}{{(9 x)}^{2}}$

Step 4

Evaluate $\frac{d}{d x} [- 18 x]$ .

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Step 4.1

Since $- 18$ is constant with respect to $x$ , the derivative of $- 18 x$ with respect to $x$ is $- 18 \frac{d}{d x} [x]$ .

$\frac{9 x (81 x^{2} - 18 \frac{d}{d x} [x]) - (27 x^{3} - 18 x) \frac{d}{d x} [9 x]}{{(9 x)}^{2}}$

Step 4.2

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

$\frac{9 x (81 x^{2} - 18 \cdot 1) - (27 x^{3} - 18 x) \frac{d}{d x} [9 x]}{{(9 x)}^{2}}$

Step 4.3

Multiply $- 18$ by $1$ .

$\frac{9 x (81 x^{2} - 18) - (27 x^{3} - 18 x) \frac{d}{d x} [9 x]}{{(9 x)}^{2}}$

Step 5

Differentiate.

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Step 5.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x$ with respect to $x$ is $9 \frac{d}{d x} [x]$ .

$\frac{9 x (81 x^{2} - 18) - (27 x^{3} - 18 x) (9 \frac{d}{d x} [x])}{{(9 x)}^{2}}$

Step 5.2

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

$\frac{9 x (81 x^{2} - 18) - (27 x^{3} - 18 x) (9 \cdot 1)}{{(9 x)}^{2}}$

Step 5.3

Multiply $9$ by $1$ .

$\frac{9 x (81 x^{2} - 18) - (27 x^{3} - 18 x) \cdot 9}{{(9 x)}^{2}}$

Step 6

Simplify.

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Step 6.1

Apply the product rule to $9 x$ .

$\frac{9 x (81 x^{2} - 18) - (27 x^{3} - 18 x) \cdot 9}{9^{2} x^{2}}$

Step 6.2

Apply the distributive property.

$\frac{9 x (81 x^{2}) + 9 x \cdot - 18 - (27 x^{3} - 18 x) \cdot 9}{9^{2} x^{2}}$

Step 6.3

Apply the distributive property.

$\frac{9 x (81 x^{2}) + 9 x \cdot - 18 + (- (27 x^{3}) - (- 18 x)) \cdot 9}{9^{2} x^{2}}$

Step 6.4

Apply the distributive property.

$\frac{9 x (81 x^{2}) + 9 x \cdot - 18 - (27 x^{3}) \cdot 9 - (- 18 x) \cdot 9}{9^{2} x^{2}}$

Step 6.5

Simplify the numerator.

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Step 6.5.1

Simplify each term.

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Step 6.5.1.1

Rewrite using the commutative property of multiplication.

$\frac{9 \cdot 81 x \cdot x^{2} + 9 x \cdot - 18 - (27 x^{3}) \cdot 9 - (- 18 x) \cdot 9}{9^{2} x^{2}}$

Step 6.5.1.2

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

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Step 6.5.1.2.1

Move $x^{2}$ .

$\frac{9 \cdot 81 (x^{2} x) + 9 x \cdot - 18 - (27 x^{3}) \cdot 9 - (- 18 x) \cdot 9}{9^{2} x^{2}}$

Step 6.5.1.2.2

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

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Step 6.5.1.2.2.1

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

$\frac{9 \cdot 81 (x^{2} x^{1}) + 9 x \cdot - 18 - (27 x^{3}) \cdot 9 - (- 18 x) \cdot 9}{9^{2} x^{2}}$

Step 6.5.1.2.2.2

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

$\frac{9 \cdot 81 x^{2 + 1} + 9 x \cdot - 18 - (27 x^{3}) \cdot 9 - (- 18 x) \cdot 9}{9^{2} x^{2}}$

Step 6.5.1.2.3

Add $2$ and $1$ .

$\frac{9 \cdot 81 x^{3} + 9 x \cdot - 18 - (27 x^{3}) \cdot 9 - (- 18 x) \cdot 9}{9^{2} x^{2}}$

Step 6.5.1.3

Multiply $9$ by $81$ .

$\frac{729 x^{3} + 9 x \cdot - 18 - (27 x^{3}) \cdot 9 - (- 18 x) \cdot 9}{9^{2} x^{2}}$

Step 6.5.1.4

Multiply $- 18$ by $9$ .

$\frac{729 x^{3} - 162 x - (27 x^{3}) \cdot 9 - (- 18 x) \cdot 9}{9^{2} x^{2}}$

Step 6.5.1.5

Multiply $27$ by $- 1$ .

$\frac{729 x^{3} - 162 x - 27 x^{3} \cdot 9 - (- 18 x) \cdot 9}{9^{2} x^{2}}$

Step 6.5.1.6

Multiply $9$ by $- 27$ .

$\frac{729 x^{3} - 162 x - 243 x^{3} - (- 18 x) \cdot 9}{9^{2} x^{2}}$

Step 6.5.1.7

Multiply $- 18$ by $- 1$ .

$\frac{729 x^{3} - 162 x - 243 x^{3} + 18 x \cdot 9}{9^{2} x^{2}}$

Step 6.5.1.8

Multiply $9$ by $18$ .

$\frac{729 x^{3} - 162 x - 243 x^{3} + 162 x}{9^{2} x^{2}}$

Step 6.5.2

Combine the opposite terms in $729 x^{3} - 162 x - 243 x^{3} + 162 x$ .

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Step 6.5.2.1

Add $- 162 x$ and $162 x$ .

$\frac{729 x^{3} - 243 x^{3} + 0}{9^{2} x^{2}}$

Step 6.5.2.2

Add $729 x^{3} - 243 x^{3}$ and $0$ .

$\frac{729 x^{3} - 243 x^{3}}{9^{2} x^{2}}$

Step 6.5.3

Subtract $243 x^{3}$ from $729 x^{3}$ .

$\frac{486 x^{3}}{9^{2} x^{2}}$

Step 6.6

Combine terms.

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Step 6.6.1

Raise $9$ to the power of $2$ .

$\frac{486 x^{3}}{81 x^{2}}$

Step 6.6.2

Cancel the common factor of $486$ and $81$ .

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Step 6.6.2.1

Factor $81$ out of $486 x^{3}$ .

$\frac{81 (6 x^{3})}{81 x^{2}}$

Step 6.6.2.2

Cancel the common factors.

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Step 6.6.2.2.1

Factor $81$ out of $81 x^{2}$ .

$\frac{81 (6 x^{3})}{81 (x^{2})}$

Step 6.6.2.2.2

Cancel the common factor.

$\frac{81 (6 x^{3})}{81 x^{2}}$

Step 6.6.2.2.3

Rewrite the expression.

$\frac{6 x^{3}}{x^{2}}$

Step 6.6.3

Cancel the common factor of $x^{3}$ and $x^{2}$ .

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Step 6.6.3.1

Factor $x^{2}$ out of $6 x^{3}$ .

$\frac{x^{2} (6 x)}{x^{2}}$

Step 6.6.3.2

Cancel the common factors.

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Step 6.6.3.2.1

Multiply by $1$ .

$\frac{x^{2} (6 x)}{x^{2} \cdot 1}$

Step 6.6.3.2.2

Cancel the common factor.

$\frac{x^{2} (6 x)}{x^{2} \cdot 1}$

Step 6.6.3.2.3

Rewrite the expression.

$\frac{6 x}{1}$

Step 6.6.3.2.4

Divide $6 x$ by $1$ .

$6 x$