Calculus Examples

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

Step 1

Since $9$ is constant with respect to $x$ , the derivative of $\frac{9 (x^{2} - 3)}{x^{3}}$ with respect to $x$ is $9 \frac{d}{d x} [\frac{x^{2} - 3}{x^{3}}]$ .

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

Step 2

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

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

Step 3

Differentiate.

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Step 3.1

Multiply the exponents in ${(x^{3})}^{2}$ .

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Step 3.1.1

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

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

Step 3.1.2

Multiply $3$ by $2$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$9 \frac{x^{3} (2 x + 0) - (x^{2} - 3) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 3.5

Add $2 x$ and $0$ .

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

Step 4

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 4.1

Move $x$ .

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

Step 4.2

Multiply $x$ by $x^{3}$ .

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Step 4.2.1

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

$9 \frac{x^{1} x^{3} \cdot 2 - (x^{2} - 3) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 4.2.2

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

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

Step 4.3

Add $1$ and $3$ .

$9 \frac{x^{4} \cdot 2 - (x^{2} - 3) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 5

Move $2$ to the left of $x^{4}$ .

$9 \frac{2 \cdot x^{4} - (x^{2} - 3) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 6

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

$9 \frac{2 x^{4} - (x^{2} - 3) (3 x^{2})}{x^{6}}$

Step 7

Simplify with factoring out.

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Step 7.1

Multiply $3$ by $- 1$ .

$9 \frac{2 x^{4} - 3 (x^{2} - 3) x^{2}}{x^{6}}$

Step 7.2

Factor $x^{2}$ out of $2 x^{4} - 3 (x^{2} - 3) x^{2}$ .

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Step 7.2.1

Factor $x^{2}$ out of $2 x^{4}$ .

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

Step 7.2.2

Factor $x^{2}$ out of $- 3 (x^{2} - 3) x^{2}$ .

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

Step 7.2.3

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

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

Step 8

Cancel the common factors.

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Step 8.1

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

$9 \frac{x^{2} (2 x^{2} - 3 (x^{2} - 3))}{x^{2} x^{4}}$

Step 8.2

Cancel the common factor.

$9 \frac{x^{2} (2 x^{2} - 3 (x^{2} - 3))}{x^{2} x^{4}}$

Step 8.3

Rewrite the expression.

$9 \frac{2 x^{2} - 3 (x^{2} - 3)}{x^{4}}$

Step 9

Combine $9$ and $\frac{2 x^{2} - 3 (x^{2} - 3)}{x^{4}}$ .

$\frac{9 (2 x^{2} - 3 (x^{2} - 3))}{x^{4}}$

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

$\frac{9 (2 x^{2} - 3 x^{2} - 3 \cdot - 3)}{x^{4}}$

Step 10.2

Apply the distributive property.

$\frac{9 (2 x^{2}) + 9 (- 3 x^{2}) + 9 (- 3 \cdot - 3)}{x^{4}}$

Step 10.3

Simplify the numerator.

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Step 10.3.1

Simplify each term.

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Step 10.3.1.1

Multiply $2$ by $9$ .

$\frac{18 x^{2} + 9 (- 3 x^{2}) + 9 (- 3 \cdot - 3)}{x^{4}}$

Step 10.3.1.2

Multiply $- 3$ by $9$ .

$\frac{18 x^{2} - 27 x^{2} + 9 (- 3 \cdot - 3)}{x^{4}}$

Step 10.3.1.3

Multiply $9 (- 3 \cdot - 3)$ .

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Step 10.3.1.3.1

Multiply $- 3$ by $- 3$ .

$\frac{18 x^{2} - 27 x^{2} + 9 \cdot 9}{x^{4}}$

Step 10.3.1.3.2

Multiply $9$ by $9$ .

$\frac{18 x^{2} - 27 x^{2} + 81}{x^{4}}$

Step 10.3.2

Subtract $27 x^{2}$ from $18 x^{2}$ .

$\frac{- 9 x^{2} + 81}{x^{4}}$

Step 10.4

Simplify the numerator.

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Step 10.4.1

Factor $9$ out of $- 9 x^{2} + 81$ .

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Step 10.4.1.1

Factor $9$ out of $- 9 x^{2}$ .

$\frac{9 (- x^{2}) + 81}{x^{4}}$

Step 10.4.1.2

Factor $9$ out of $81$ .

$\frac{9 (- x^{2}) + 9 (9)}{x^{4}}$

Step 10.4.1.3

Factor $9$ out of $9 (- x^{2}) + 9 (9)$ .

$\frac{9 (- x^{2} + 9)}{x^{4}}$

Step 10.4.2

Rewrite $9$ as $3^{2}$ .

$\frac{9 (- x^{2} + 3^{2})}{x^{4}}$

Step 10.4.3

Reorder $- x^{2}$ and $3^{2}$ .

$\frac{9 (3^{2} - x^{2})}{x^{4}}$

Step 10.4.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3$ and $b = x$ .

$\frac{9 (3 + x) (3 - x)}{x^{4}}$