Calculus Examples

$\ln (| x^{2} - 1 |)$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = | x^{2} - 1 |$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $| x^{2} - 1 |$ .

$\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [| x^{2} - 1 |]$

Step 1.2

The derivative of $\ln (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1}}$ .

$\frac{1}{u_{1}} \frac{d}{d x} [| x^{2} - 1 |]$

Step 1.3

Replace all occurrences of $u_{1}$ with $| x^{2} - 1 |$ .

$\frac{1}{| x^{2} - 1 |} \frac{d}{d x} [| x^{2} - 1 |]$

Step 2

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 $g (x) = x^{2} - 1$ .

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Step 2.1

To apply the Chain Rule, set $u_{2}$ as $x^{2} - 1$ .

$\frac{1}{| x^{2} - 1 |} (\frac{d}{d u_{2}} [| u_{2} |] \frac{d}{d x} [x^{2} - 1])$

Step 2.2

The derivative of $| u_{2} |$ with respect to $u_{2}$ is $\frac{u_{2}}{| u_{2} |}$ .

$\frac{1}{| x^{2} - 1 |} (\frac{u_{2}}{| u_{2} |} \frac{d}{d x} [x^{2} - 1])$

Step 2.3

Replace all occurrences of $u_{2}$ with $x^{2} - 1$ .

$\frac{1}{| x^{2} - 1 |} (\frac{x^{2} - 1}{| x^{2} - 1 |} \frac{d}{d x} [x^{2} - 1])$

Step 3

Multiply $\frac{x^{2} - 1}{| x^{2} - 1 |}$ by $\frac{1}{| x^{2} - 1 |}$ .

$\frac{x^{2} - 1}{| x^{2} - 1 | | x^{2} - 1 |} \frac{d}{d x} [x^{2} - 1]$

Step 4

To multiply absolute values, multiply the terms inside each absolute value.

$\frac{x^{2} - 1}{| (x^{2} - 1) (x^{2} - 1) |} \frac{d}{d x} [x^{2} - 1]$

Step 5

Raise $x^{2} - 1$ to the power of $1$ .

$\frac{x^{2} - 1}{| {(x^{2} - 1)}^{1} (x^{2} - 1) |} \frac{d}{d x} [x^{2} - 1]$

Step 6

Raise $x^{2} - 1$ to the power of $1$ .

$\frac{x^{2} - 1}{| {(x^{2} - 1)}^{1} {(x^{2} - 1)}^{1} |} \frac{d}{d x} [x^{2} - 1]$

Step 7

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

$\frac{x^{2} - 1}{| {(x^{2} - 1)}^{1 + 1} |} \frac{d}{d x} [x^{2} - 1]$

Step 8

Add $1$ and $1$ .

$\frac{x^{2} - 1}{| {(x^{2} - 1)}^{2} |} \frac{d}{d x} [x^{2} - 1]$

Step 9

By the Sum Rule, the derivative of $x^{2} - 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]$ .

$\frac{x^{2} - 1}{| {(x^{2} - 1)}^{2} |} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1])$

Step 10

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

$\frac{x^{2} - 1}{| {(x^{2} - 1)}^{2} |} (2 x + \frac{d}{d x} [- 1])$

Step 11

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$\frac{x^{2} - 1}{| {(x^{2} - 1)}^{2} |} (2 x + 0)$

Step 12

Combine fractions.

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Step 12.1

Add $2 x$ and $0$ .

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

Step 12.2

Combine $2$ and $\frac{x^{2} - 1}{| {(x^{2} - 1)}^{2} |}$ .

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

Step 12.3

Combine $\frac{2 (x^{2} - 1)}{| {(x^{2} - 1)}^{2} |}$ and $x$ .

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

Step 13

Simplify.

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Step 13.1

Apply the distributive property.

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

Step 13.2

Apply the distributive property.

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

Step 13.3

Simplify each term.

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Step 13.3.1

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

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Step 13.3.1.1

Move $x$ .

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

Step 13.3.1.2

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

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Step 13.3.1.2.1

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

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

Step 13.3.1.2.2

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

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

Step 13.3.1.3

Add $1$ and $2$ .

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

Step 13.3.2

Multiply $2$ by $- 1$ .

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