Calculus Examples

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

Step 1

Find the second derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

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

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

Step 1.1.2

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

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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

Step 1.1.3

Differentiate.

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Step 1.1.3.1

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

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

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

Step 1.1.3.4

Simplify the expression.

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Step 1.1.3.4.1

Add $2 x$ and $0$ .

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

Step 1.1.3.4.2

Multiply $2$ by $- 1$ .

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

Step 1.1.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.1.5

Combine terms.

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Step 1.1.5.1

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

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

Step 1.1.5.2

Move the negative in front of the fraction.

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

Step 1.1.5.3

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

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

Step 1.1.5.4

Move $2$ to the left of $x$ .

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

Step 1.2

Find the second derivative.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

Differentiate using the Power Rule.

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Step 1.2.3.1

Multiply the exponents in ${({(x^{2} - 1)}^{2})}^{2}$ .

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Step 1.2.3.1.1

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

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

Step 1.2.3.1.2

Multiply $2$ by $2$ .

$f'' (x) = - 2 \frac{{(x^{2} - 1)}^{2} \frac{d}{d x} (x) - x \frac{d}{d x} {(x^{2} - 1)}^{2}}{{(x^{2} - 1)}^{4}}$

Step 1.2.3.2

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

$f'' (x) = - 2 \frac{{(x^{2} - 1)}^{2} \cdot 1 - x \frac{d}{d x} {(x^{2} - 1)}^{2}}{{(x^{2} - 1)}^{4}}$

Step 1.2.3.3

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

$f'' (x) = - 2 \frac{{(x^{2} - 1)}^{2} - x \frac{d}{d x} {(x^{2} - 1)}^{2}}{{(x^{2} - 1)}^{4}}$

Step 1.2.4

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

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Step 1.2.4.1

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

$f'' (x) = - 2 \frac{{(x^{2} - 1)}^{2} - x (\frac{d}{d u} (u^{2}) \frac{d}{d x} (x^{2} - 1))}{{(x^{2} - 1)}^{4}}$

Step 1.2.4.2

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

$f'' (x) = - 2 \frac{{(x^{2} - 1)}^{2} - x (2 u \frac{d}{d x} (x^{2} - 1))}{{(x^{2} - 1)}^{4}}$

Step 1.2.4.3

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

$f'' (x) = - 2 \frac{{(x^{2} - 1)}^{2} - x (2 (x^{2} - 1) \frac{d}{d x} (x^{2} - 1))}{{(x^{2} - 1)}^{4}}$

Step 1.2.5

Simplify with factoring out.

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Step 1.2.5.1

Multiply $2$ by $- 1$ .

$f'' (x) = - 2 \frac{{(x^{2} - 1)}^{2} - 2 x ((x^{2} - 1) \frac{d}{d x} (x^{2} - 1))}{{(x^{2} - 1)}^{4}}$

Step 1.2.5.2

Factor $x^{2} - 1$ out of ${(x^{2} - 1)}^{2} - 2 x ((x^{2} - 1) \frac{d}{d x} [x^{2} - 1])$ .

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Step 1.2.5.2.1

Factor $x^{2} - 1$ out of ${(x^{2} - 1)}^{2}$ .

$f'' (x) = - 2 \frac{(x^{2} - 1) (x^{2} - 1) - 2 x ((x^{2} - 1) \frac{d}{d x} (x^{2} - 1))}{{(x^{2} - 1)}^{4}}$

Step 1.2.5.2.2

Factor $x^{2} - 1$ out of $- 2 x ((x^{2} - 1) \frac{d}{d x} [x^{2} - 1])$ .

$f'' (x) = - 2 \frac{(x^{2} - 1) (x^{2} - 1) + (x^{2} - 1) (- 2 x (\frac{d}{d x} (x^{2} - 1)))}{{(x^{2} - 1)}^{4}}$

Step 1.2.5.2.3

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

$f'' (x) = - 2 \frac{(x^{2} - 1) (x^{2} - 1 - 2 x (\frac{d}{d x} (x^{2} - 1)))}{{(x^{2} - 1)}^{4}}$

Step 1.2.6

Cancel the common factors.

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Step 1.2.6.1

Factor $x^{2} - 1$ out of ${(x^{2} - 1)}^{4}$ .

$f'' (x) = - 2 \frac{(x^{2} - 1) (x^{2} - 1 - 2 x (\frac{d}{d x} (x^{2} - 1)))}{(x^{2} - 1) {(x^{2} - 1)}^{3}}$

Step 1.2.6.2

Cancel the common factor.

$f'' (x) = - 2 \frac{(x^{2} - 1) (x^{2} - 1 - 2 x (\frac{d}{d x} (x^{2} - 1)))}{(x^{2} - 1) {(x^{2} - 1)}^{3}}$

Step 1.2.6.3

Rewrite the expression.

$f'' (x) = - 2 \frac{x^{2} - 1 - 2 x (\frac{d}{d x} (x^{2} - 1))}{{(x^{2} - 1)}^{3}}$

Step 1.2.7

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

$f'' (x) = - 2 \frac{x^{2} - 1 - 2 x (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 1))}{{(x^{2} - 1)}^{3}}$

Step 1.2.8

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

$f'' (x) = - 2 \frac{x^{2} - 1 - 2 x (2 x + \frac{d}{d x} (- 1))}{{(x^{2} - 1)}^{3}}$

Step 1.2.9

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

$f'' (x) = - 2 \frac{x^{2} - 1 - 2 x (2 x + 0)}{{(x^{2} - 1)}^{3}}$

Step 1.2.10

Simplify the expression.

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Step 1.2.10.1

Add $2 x$ and $0$ .

$f'' (x) = - 2 \frac{x^{2} - 1 - 2 x (2 x)}{{(x^{2} - 1)}^{3}}$

Step 1.2.10.2

Multiply $2$ by $- 2$ .

$f'' (x) = - 2 \frac{x^{2} - 1 - 4 x \cdot x}{{(x^{2} - 1)}^{3}}$

Step 1.2.11

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

$f'' (x) = - 2 \frac{x^{2} - 1 - 4 (x \cdot x)}{{(x^{2} - 1)}^{3}}$

Step 1.2.12

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

$f'' (x) = - 2 \frac{x^{2} - 1 - 4 (x \cdot x)}{{(x^{2} - 1)}^{3}}$

Step 1.2.13

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

$f'' (x) = - 2 \frac{x^{2} - 1 - 4 x^{1 + 1}}{{(x^{2} - 1)}^{3}}$

Step 1.2.14

Add $1$ and $1$ .

$f'' (x) = - 2 \frac{x^{2} - 1 - 4 x^{2}}{{(x^{2} - 1)}^{3}}$

Step 1.2.15

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

$f'' (x) = - 2 \frac{- 3 x^{2} - 1}{{(x^{2} - 1)}^{3}}$

Step 1.2.16

Combine $- 2$ and $\frac{- 3 x^{2} - 1}{{(x^{2} - 1)}^{3}}$ .

$f'' (x) = \frac{- 2 (- 3 x^{2} - 1)}{{(x^{2} - 1)}^{3}}$

Step 1.2.17

Move the negative in front of the fraction.

$f'' (x) = - \frac{(2) (- 3 x^{2} - 1)}{{(x^{2} - 1)}^{3}}$

Step 1.2.18

Simplify.

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Step 1.2.18.1

Apply the distributive property.

$f'' (x) = - \frac{2 (- 3 x^{2}) + 2 \cdot - 1}{{(x^{2} - 1)}^{3}}$

Step 1.2.18.2

Simplify each term.

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Step 1.2.18.2.1

Multiply $- 3$ by $2$ .

$f'' (x) = - \frac{- 6 x^{2} + 2 \cdot - 1}{{(x^{2} - 1)}^{3}}$

Step 1.2.18.2.2

Multiply $2$ by $- 1$ .

$f'' (x) = - \frac{- 6 x^{2} - 2}{{(x^{2} - 1)}^{3}}$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $- \frac{- 6 x^{2} - 2}{{(x^{2} - 1)}^{3}}$ .

$- \frac{- 6 x^{2} - 2}{{(x^{2} - 1)}^{3}}$

Step 2

Set the second derivative equal to $0$ then solve the equation $- \frac{- 6 x^{2} - 2}{{(x^{2} - 1)}^{3}} = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$- \frac{- 6 x^{2} - 2}{{(x^{2} - 1)}^{3}} = 0$

Step 2.2

Set the numerator equal to zero.

$- 6 x^{2} - 2 = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Add $2$ to both sides of the equation.

$- 6 x^{2} = 2$

Step 2.3.2

Divide each term in $- 6 x^{2} = 2$ by $- 6$ and simplify.

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Step 2.3.2.1

Divide each term in $- 6 x^{2} = 2$ by $- 6$ .

$\frac{- 6 x^{2}}{- 6} = \frac{2}{- 6}$

Step 2.3.2.2

Simplify the left side.

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Step 2.3.2.2.1

Cancel the common factor of $- 6$ .

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Step 2.3.2.2.1.1

Cancel the common factor.

$\frac{- 6 x^{2}}{- 6} = \frac{2}{- 6}$

Step 2.3.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{2}{- 6}$

Step 2.3.2.3

Simplify the right side.

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Step 2.3.2.3.1

Cancel the common factor of $2$ and $- 6$ .

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Step 2.3.2.3.1.1

Factor $2$ out of $2$ .

$x^{2} = \frac{2 (1)}{- 6}$

Step 2.3.2.3.1.2

Cancel the common factors.

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Step 2.3.2.3.1.2.1

Factor $2$ out of $- 6$ .

$x^{2} = \frac{2 \cdot 1}{2 \cdot - 3}$

Step 2.3.2.3.1.2.2

Cancel the common factor.

$x^{2} = \frac{2 \cdot 1}{2 \cdot - 3}$

Step 2.3.2.3.1.2.3

Rewrite the expression.

$x^{2} = \frac{1}{- 3}$

Step 2.3.2.3.2

Move the negative in front of the fraction.

$x^{2} = - \frac{1}{3}$

Step 2.3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- \frac{1}{3}}$

Step 2.3.4

Simplify $\pm \sqrt{- \frac{1}{3}}$ .

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Step 2.3.4.1

Rewrite $- \frac{1}{3}$ as $i^{2} \frac{1^{2}}{3}$ .

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Step 2.3.4.1.1

Rewrite $- 1$ as $i^{2}$ .

$x = \pm \sqrt{i^{2} \frac{1}{3}}$

Step 2.3.4.1.2

Rewrite $1$ as $1^{2}$ .

$x = \pm \sqrt{i^{2} \frac{1^{2}}{3}}$

Step 2.3.4.2

Pull terms out from under the radical.

$x = \pm i \sqrt{\frac{1^{2}}{3}}$

Step 2.3.4.3

One to any power is one.

$x = \pm i \sqrt{\frac{1}{3}}$

Step 2.3.4.4

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

$x = \pm i \frac{\sqrt{1}}{\sqrt{3}}$

Step 2.3.4.5

Any root of $1$ is $1$ .

$x = \pm i \frac{1}{\sqrt{3}}$

Step 2.3.4.6

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm i (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 2.3.4.7

Combine and simplify the denominator.

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Step 2.3.4.7.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm i \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 2.3.4.7.2

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 2.3.4.7.3

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 2.3.4.7.4

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

$x = \pm i \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 2.3.4.7.5

Add $1$ and $1$ .

$x = \pm i \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Step 2.3.4.7.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.3.4.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = \pm i \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 2.3.4.7.6.2

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

$x = \pm i \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 2.3.4.7.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \pm i \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.3.4.7.6.4

Cancel the common factor of $2$ .

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Step 2.3.4.7.6.4.1

Cancel the common factor.

$x = \pm i \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.3.4.7.6.4.2

Rewrite the expression.

$x = \pm i \frac{\sqrt{3}}{3^{1}}$

Step 2.3.4.7.6.5

Evaluate the exponent.

$x = \pm i \frac{\sqrt{3}}{3}$

Step 2.3.4.8

Combine $i$ and $\frac{\sqrt{3}}{3}$ .

$x = \pm \frac{i \sqrt{3}}{3}$

Step 2.3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{i \sqrt{3}}{3}$

Step 2.3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{i \sqrt{3}}{3}$

Step 2.3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{i \sqrt{3}}{3}, - \frac{i \sqrt{3}}{3}$

Step 3

No values found that can make the second derivative equal to $0$ .

No Inflection Points