Calculus Examples

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

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

Move $3$ to the left of $x^{2} - 9$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$\frac{3 (x^{2} - 9) x^{2} - x^{3} (2 x + 0)}{{(x^{2} - 9)}^{2}}$

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $2 x$ and $0$ .

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

Step 2.6.2

Multiply $2$ by $- 1$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Add $1$ and $3$ .

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

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Simplify the numerator.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

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

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Step 6.3.1.1.1

Move $x^{2}$ .

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

Step 6.3.1.1.2

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

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

Step 6.3.1.1.3

Add $2$ and $2$ .

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

Step 6.3.1.2

Multiply $3$ by $- 9$ .

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

Step 6.3.2

Subtract $2 x^{4}$ from $3 x^{4}$ .

$\frac{x^{4} - 27 x^{2}}{{(x^{2} - 9)}^{2}}$

Step 6.4

Factor $x^{2}$ out of $x^{4} - 27 x^{2}$ .

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Step 6.4.1

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

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

Step 6.4.2

Factor $x^{2}$ out of $- 27 x^{2}$ .

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

Step 6.4.3

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} \cdot - 27$ .

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

Step 6.5

Simplify the denominator.

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Step 6.5.1

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

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

Step 6.5.2

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

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

Step 6.5.3

Apply the product rule to $(x + 3) (x - 3)$ .

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