Calculus Examples

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

Step 1

Find the first derivative of the function.

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Step 1.1

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 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

Combine fractions.

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Step 1.2.4.1

Add $2 x$ and $0$ .

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

Step 1.2.4.2

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

$\frac{2}{x^{2} - 1} x$

Step 1.2.4.3

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

$\frac{2 x}{x^{2} - 1}$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2 x}{x^{2} - 1}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{x}{x^{2} - 1}]$ .

$f'' (x) = 2 \frac{d}{d x} (\frac{x}{x^{2} - 1})$

Step 2.2

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

$f'' (x) = 2 (\frac{(x^{2} - 1) \frac{d}{d x} (x) - x \frac{d}{d x} (x^{2} - 1)}{{(x^{2} - 1)}^{2}})$

Step 2.3

Differentiate.

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Step 2.3.1

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

$f'' (x) = 2 (\frac{(x^{2} - 1) \cdot 1 - x \frac{d}{d x} (x^{2} - 1)}{{(x^{2} - 1)}^{2}})$

Step 2.3.2

Multiply $x^{2} - 1$ by $1$ .

$f'' (x) = 2 (\frac{x^{2} - 1 - x \frac{d}{d x} (x^{2} - 1)}{{(x^{2} - 1)}^{2}})$

Step 2.3.3

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

$f'' (x) = 2 (\frac{x^{2} - 1 - x (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 1))}{{(x^{2} - 1)}^{2}})$

Step 2.3.4

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

$f'' (x) = 2 (\frac{x^{2} - 1 - x (2 x + \frac{d}{d x} (- 1))}{{(x^{2} - 1)}^{2}})$

Step 2.3.5

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

$f'' (x) = 2 (\frac{x^{2} - 1 - x (2 x + 0)}{{(x^{2} - 1)}^{2}})$

Step 2.3.6

Simplify the expression.

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Step 2.3.6.1

Add $2 x$ and $0$ .

$f'' (x) = 2 (\frac{x^{2} - 1 - x (2 x)}{{(x^{2} - 1)}^{2}})$

Step 2.3.6.2

Multiply $2$ by $- 1$ .

$f'' (x) = 2 (\frac{x^{2} - 1 - 2 x \cdot x}{{(x^{2} - 1)}^{2}})$

Step 2.4

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

$f'' (x) = 2 (\frac{x^{2} - 1 - 2 (x \cdot x)}{{(x^{2} - 1)}^{2}})$

Step 2.5

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

$f'' (x) = 2 (\frac{x^{2} - 1 - 2 (x \cdot x)}{{(x^{2} - 1)}^{2}})$

Step 2.6

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

$f'' (x) = 2 (\frac{x^{2} - 1 - 2 x^{1 + 1}}{{(x^{2} - 1)}^{2}})$

Step 2.7

Add $1$ and $1$ .

$f'' (x) = 2 (\frac{x^{2} - 1 - 2 x^{2}}{{(x^{2} - 1)}^{2}})$

Step 2.8

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

$f'' (x) = 2 (\frac{- x^{2} - 1}{{(x^{2} - 1)}^{2}})$

Step 2.9

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

$f'' (x) = \frac{2 (- x^{2} - 1)}{{(x^{2} - 1)}^{2}}$

Step 2.10

Simplify.

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Step 2.10.1

Apply the distributive property.

$f'' (x) = \frac{2 (- x^{2}) + 2 \cdot - 1}{{(x^{2} - 1)}^{2}}$

Step 2.10.2

Simplify each term.

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Step 2.10.2.1

Multiply $- 1$ by $2$ .

$f'' (x) = \frac{- 2 x^{2} + 2 \cdot - 1}{{(x^{2} - 1)}^{2}}$

Step 2.10.2.2

Multiply $2$ by $- 1$ .

$f'' (x) = \frac{- 2 x^{2} - 2}{{(x^{2} - 1)}^{2}}$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$\frac{2 x}{x^{2} - 1} = 0$

Step 4

Since there is no value of $x$ that makes the first derivative equal to $0$ , there are no local extrema.

No Local Extrema

Step 5

No Local Extrema

Step 6