Calculus Examples

$f (x) = | x^{2} - 9 |$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = | x |$ and $g (x) = x^{2} - 9$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $x^{2} - 9$ .

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

Step 1.2

The derivative of $| u |$ with respect to $u$ is $\frac{u}{| u |}$ .

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

Step 1.3

Replace all occurrences of $u$ with $x^{2} - 9$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Combine fractions.

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Step 2.4.1

Add $2 x$ and $0$ .

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

Step 2.4.2

Combine $2$ and $\frac{x^{2} - 9}{| x^{2} - 9 |}$ .

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

Step 2.4.3

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

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

$\frac{2 x^{2} x + 2 \cdot - 9 x}{| x^{2} - 9 |}$

Step 3.3

Simplify each term.

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Step 3.3.1

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

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Step 3.3.1.1

Move $x$ .

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

Step 3.3.1.2

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

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Step 3.3.1.2.1

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

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

Step 3.3.1.2.2

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

$\frac{2 x^{1 + 2} + 2 \cdot - 9 x}{| x^{2} - 9 |}$

Step 3.3.1.3

Add $1$ and $2$ .

$\frac{2 x^{3} + 2 \cdot - 9 x}{| x^{2} - 9 |}$

Step 3.3.2

Multiply $2$ by $- 9$ .

$\frac{2 x^{3} - 18 x}{| x^{2} - 9 |}$