Calculus Examples

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

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 $f (x) = 9 x^{3} + 3 x^{2}$ and $g (x) = x$ .

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

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $9 x^{3} + 3 x^{2}$ with respect to $x$ is $\frac{d}{d x} [9 x^{3}] + \frac{d}{d x} [3 x^{2}]$ .

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

Step 2.2

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{3}$ with respect to $x$ is $9 \frac{d}{d x} [x^{3}]$ .

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

Step 2.3

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

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

Step 2.4

Multiply $3$ by $9$ .

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

Step 2.5

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

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

Step 2.6

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

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

Step 2.7

Multiply $2$ by $3$ .

$\frac{x (27 x^{2} + 6 x) - (9 x^{3} + 3 x^{2}) \frac{d}{d x} [x]}{x^{2}}$

Step 2.8

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

$\frac{x (27 x^{2} + 6 x) - (9 x^{3} + 3 x^{2}) \cdot 1}{x^{2}}$

Step 2.9

Multiply $- 1$ by $1$ .

$\frac{x (27 x^{2} + 6 x) - (9 x^{3} + 3 x^{2})}{x^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{x (27 x^{2}) + x (6 x) - (9 x^{3} + 3 x^{2})}{x^{2}}$

Step 3.2

Apply the distributive property.

$\frac{x (27 x^{2}) + x (6 x) - (9 x^{3}) - (3 x^{2})}{x^{2}}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Rewrite using the commutative property of multiplication.

$\frac{27 x \cdot x^{2} + x (6 x) - (9 x^{3}) - (3 x^{2})}{x^{2}}$

Step 3.3.1.2

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

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Step 3.3.1.2.1

Move $x^{2}$ .

$\frac{27 (x^{2} x) + x (6 x) - (9 x^{3}) - (3 x^{2})}{x^{2}}$

Step 3.3.1.2.2

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

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Step 3.3.1.2.2.1

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

$\frac{27 (x^{2} x^{1}) + x (6 x) - (9 x^{3}) - (3 x^{2})}{x^{2}}$

Step 3.3.1.2.2.2

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

$\frac{27 x^{2 + 1} + x (6 x) - (9 x^{3}) - (3 x^{2})}{x^{2}}$

Step 3.3.1.2.3

Add $2$ and $1$ .

$\frac{27 x^{3} + x (6 x) - (9 x^{3}) - (3 x^{2})}{x^{2}}$

Step 3.3.1.3

Rewrite using the commutative property of multiplication.

$\frac{27 x^{3} + 6 x \cdot x - (9 x^{3}) - (3 x^{2})}{x^{2}}$

Step 3.3.1.4

Multiply $x$ by $x$ by adding the exponents.

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Step 3.3.1.4.1

Move $x$ .

$\frac{27 x^{3} + 6 (x \cdot x) - (9 x^{3}) - (3 x^{2})}{x^{2}}$

Step 3.3.1.4.2

Multiply $x$ by $x$ .

$\frac{27 x^{3} + 6 x^{2} - (9 x^{3}) - (3 x^{2})}{x^{2}}$

Step 3.3.1.5

Multiply $9$ by $- 1$ .

$\frac{27 x^{3} + 6 x^{2} - 9 x^{3} - (3 x^{2})}{x^{2}}$

Step 3.3.1.6

Multiply $3$ by $- 1$ .

$\frac{27 x^{3} + 6 x^{2} - 9 x^{3} - 3 x^{2}}{x^{2}}$

Step 3.3.2

Subtract $9 x^{3}$ from $27 x^{3}$ .

$\frac{18 x^{3} + 6 x^{2} - 3 x^{2}}{x^{2}}$

Step 3.3.3

Subtract $3 x^{2}$ from $6 x^{2}$ .

$\frac{18 x^{3} + 3 x^{2}}{x^{2}}$

Step 3.4

Factor $3 x^{2}$ out of $18 x^{3} + 3 x^{2}$ .

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Step 3.4.1

Factor $3 x^{2}$ out of $18 x^{3}$ .

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

Step 3.4.2

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

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

Step 3.4.3

Factor $3 x^{2}$ out of $3 x^{2} (6 x) + 3 x^{2} (1)$ .

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

Step 3.5

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

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Step 3.5.1

Cancel the common factor.

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

Step 3.5.2

Divide $3 (6 x + 1)$ by $1$ .

$3 (6 x + 1)$

Step 3.6

Apply the distributive property.

$3 (6 x) + 3 \cdot 1$

Step 3.7

Multiply $6$ by $3$ .

$18 x + 3 \cdot 1$

Step 3.8

Multiply $3$ by $1$ .

$18 x + 3$