Calculus Examples

$y = \frac{3}{x^{2} - 9}$

Step 1

Write $y = \frac{3}{x^{2} - 9}$ as a function.

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

Step 2

Find the second derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

Differentiate using the Constant Multiple Rule.

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Step 2.1.1.1

Since $3$ is constant with respect to $x$ , the derivative of $\frac{3}{x^{2} - 9}$ with respect to $x$ is $3 \frac{d}{d x} [\frac{1}{x^{2} - 9}]$ .

$f' (x) = 3 \frac{d}{d x} (\frac{1}{x^{2} - 9})$

Step 2.1.1.2

Rewrite $\frac{1}{x^{2} - 9}$ as ${(x^{2} - 9)}^{- 1}$ .

$f' (x) = 3 \frac{d}{d x} ({(x^{2} - 9)}^{- 1})$

Step 2.1.2

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

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Step 2.1.2.1

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

$f' (x) = 3 (\frac{d}{d u} (u^{- 1}) \frac{d}{d x} (x^{2} - 9))$

Step 2.1.2.2

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

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

Step 2.1.2.3

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

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

Step 2.1.3

Differentiate.

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Step 2.1.3.1

Multiply $- 1$ by $3$ .

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

Step 2.1.3.2

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

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

Step 2.1.3.3

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

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

Step 2.1.3.4

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

$f' (x) = - 3 {(x^{2} - 9)}^{- 2} (2 x + 0)$

Step 2.1.3.5

Simplify the expression.

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Step 2.1.3.5.1

Add $2 x$ and $0$ .

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

Step 2.1.3.5.2

Multiply $2$ by $- 3$ .

$f' (x) = - 6 {(x^{2} - 9)}^{- 2} x$

Step 2.1.4

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

$f' (x) = - 6 \frac{1}{{(x^{2} - 9)}^{2}} \cdot x$

Step 2.1.5

Combine terms.

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Step 2.1.5.1

Combine $- 6$ and $\frac{1}{{(x^{2} - 9)}^{2}}$ .

$f' (x) = \frac{- 6}{{(x^{2} - 9)}^{2}} \cdot x$

Step 2.1.5.2

Move the negative in front of the fraction.

$f' (x) = - \frac{6}{{(x^{2} - 9)}^{2}} \cdot x$

Step 2.1.5.3

Combine $x$ and $\frac{6}{{(x^{2} - 9)}^{2}}$ .

$f' (x) = - \frac{x \cdot 6}{{(x^{2} - 9)}^{2}}$

Step 2.1.5.4

Move $6$ to the left of $x$ .

$f' (x) = - \frac{6 x}{{(x^{2} - 9)}^{2}}$

Step 2.2

Find the second derivative.

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Step 2.2.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- \frac{6 x}{{(x^{2} - 9)}^{2}}$ with respect to $x$ is $- 6 \frac{d}{d x} [\frac{x}{{(x^{2} - 9)}^{2}}]$ .

$f'' (x) = - 6 \frac{d}{d x} \frac{x}{{(x^{2} - 9)}^{2}}$

Step 2.2.2

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

$f'' (x) = - 6 \frac{{(x^{2} - 9)}^{2} \frac{d}{d x} (x) - x \frac{d}{d x} {(x^{2} - 9)}^{2}}{{({(x^{2} - 9)}^{2})}^{2}}$

Step 2.2.3

Differentiate using the Power Rule.

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Step 2.2.3.1

Multiply the exponents in ${({(x^{2} - 9)}^{2})}^{2}$ .

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Step 2.2.3.1.1

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

$f'' (x) = - 6 \frac{{(x^{2} - 9)}^{2} \frac{d}{d x} (x) - x \frac{d}{d x} {(x^{2} - 9)}^{2}}{{(x^{2} - 9)}^{2 \cdot 2}}$

Step 2.2.3.1.2

Multiply $2$ by $2$ .

$f'' (x) = - 6 \frac{{(x^{2} - 9)}^{2} \frac{d}{d x} (x) - x \frac{d}{d x} {(x^{2} - 9)}^{2}}{{(x^{2} - 9)}^{4}}$

Step 2.2.3.2

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

$f'' (x) = - 6 \frac{{(x^{2} - 9)}^{2} \cdot 1 - x \frac{d}{d x} {(x^{2} - 9)}^{2}}{{(x^{2} - 9)}^{4}}$

Step 2.2.3.3

Multiply ${(x^{2} - 9)}^{2}$ by $1$ .

$f'' (x) = - 6 \frac{{(x^{2} - 9)}^{2} - x \frac{d}{d x} {(x^{2} - 9)}^{2}}{{(x^{2} - 9)}^{4}}$

Step 2.2.4

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

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Step 2.2.4.1

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

$f'' (x) = - 6 \frac{{(x^{2} - 9)}^{2} - x (\frac{d}{d u} (u^{2}) \frac{d}{d x} (x^{2} - 9))}{{(x^{2} - 9)}^{4}}$

Step 2.2.4.2

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

$f'' (x) = - 6 \frac{{(x^{2} - 9)}^{2} - x (2 u \frac{d}{d x} (x^{2} - 9))}{{(x^{2} - 9)}^{4}}$

Step 2.2.4.3

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

$f'' (x) = - 6 \frac{{(x^{2} - 9)}^{2} - x (2 (x^{2} - 9) \frac{d}{d x} (x^{2} - 9))}{{(x^{2} - 9)}^{4}}$

Step 2.2.5

Simplify with factoring out.

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Step 2.2.5.1

Multiply $2$ by $- 1$ .

$f'' (x) = - 6 \frac{{(x^{2} - 9)}^{2} - 2 x ((x^{2} - 9) \frac{d}{d x} (x^{2} - 9))}{{(x^{2} - 9)}^{4}}$

Step 2.2.5.2

Factor $x^{2} - 9$ out of ${(x^{2} - 9)}^{2} - 2 x ((x^{2} - 9) \frac{d}{d x} [x^{2} - 9])$ .

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Step 2.2.5.2.1

Factor $x^{2} - 9$ out of ${(x^{2} - 9)}^{2}$ .

$f'' (x) = - 6 \frac{(x^{2} - 9) (x^{2} - 9) - 2 x ((x^{2} - 9) \frac{d}{d x} (x^{2} - 9))}{{(x^{2} - 9)}^{4}}$

Step 2.2.5.2.2

Factor $x^{2} - 9$ out of $- 2 x ((x^{2} - 9) \frac{d}{d x} [x^{2} - 9])$ .

$f'' (x) = - 6 \frac{(x^{2} - 9) (x^{2} - 9) + (x^{2} - 9) (- 2 x (\frac{d}{d x} (x^{2} - 9)))}{{(x^{2} - 9)}^{4}}$

Step 2.2.5.2.3

Factor $x^{2} - 9$ out of $(x^{2} - 9) (x^{2} - 9) + (x^{2} - 9) (- 2 x (\frac{d}{d x} [x^{2} - 9]))$ .

$f'' (x) = - 6 \frac{(x^{2} - 9) (x^{2} - 9 - 2 x (\frac{d}{d x} (x^{2} - 9)))}{{(x^{2} - 9)}^{4}}$

Step 2.2.6

Cancel the common factors.

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Step 2.2.6.1

Factor $x^{2} - 9$ out of ${(x^{2} - 9)}^{4}$ .

$f'' (x) = - 6 \frac{(x^{2} - 9) (x^{2} - 9 - 2 x (\frac{d}{d x} (x^{2} - 9)))}{(x^{2} - 9) {(x^{2} - 9)}^{3}}$

Step 2.2.6.2

Cancel the common factor.

$f'' (x) = - 6 \frac{(x^{2} - 9) (x^{2} - 9 - 2 x (\frac{d}{d x} (x^{2} - 9)))}{(x^{2} - 9) {(x^{2} - 9)}^{3}}$

Step 2.2.6.3

Rewrite the expression.

$f'' (x) = - 6 \frac{x^{2} - 9 - 2 x (\frac{d}{d x} (x^{2} - 9))}{{(x^{2} - 9)}^{3}}$

Step 2.2.7

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

$f'' (x) = - 6 \frac{x^{2} - 9 - 2 x (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 9))}{{(x^{2} - 9)}^{3}}$

Step 2.2.8

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

$f'' (x) = - 6 \frac{x^{2} - 9 - 2 x (2 x + \frac{d}{d x} (- 9))}{{(x^{2} - 9)}^{3}}$

Step 2.2.9

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

$f'' (x) = - 6 \frac{x^{2} - 9 - 2 x (2 x + 0)}{{(x^{2} - 9)}^{3}}$

Step 2.2.10

Simplify the expression.

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Step 2.2.10.1

Add $2 x$ and $0$ .

$f'' (x) = - 6 \frac{x^{2} - 9 - 2 x (2 x)}{{(x^{2} - 9)}^{3}}$

Step 2.2.10.2

Multiply $2$ by $- 2$ .

$f'' (x) = - 6 \frac{x^{2} - 9 - 4 x \cdot x}{{(x^{2} - 9)}^{3}}$

Step 2.2.11

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

$f'' (x) = - 6 \frac{x^{2} - 9 - 4 (x \cdot x)}{{(x^{2} - 9)}^{3}}$

Step 2.2.12

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

$f'' (x) = - 6 \frac{x^{2} - 9 - 4 (x \cdot x)}{{(x^{2} - 9)}^{3}}$

Step 2.2.13

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

$f'' (x) = - 6 \frac{x^{2} - 9 - 4 x^{1 + 1}}{{(x^{2} - 9)}^{3}}$

Step 2.2.14

Add $1$ and $1$ .

$f'' (x) = - 6 \frac{x^{2} - 9 - 4 x^{2}}{{(x^{2} - 9)}^{3}}$

Step 2.2.15

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

$f'' (x) = - 6 \frac{- 3 x^{2} - 9}{{(x^{2} - 9)}^{3}}$

Step 2.2.16

Combine $- 6$ and $\frac{- 3 x^{2} - 9}{{(x^{2} - 9)}^{3}}$ .

$f'' (x) = \frac{- 6 (- 3 x^{2} - 9)}{{(x^{2} - 9)}^{3}}$

Step 2.2.17

Move the negative in front of the fraction.

$f'' (x) = - \frac{(6) (- 3 x^{2} - 9)}{{(x^{2} - 9)}^{3}}$

Step 2.2.18

Simplify.

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Step 2.2.18.1

Apply the distributive property.

$f'' (x) = - \frac{6 (- 3 x^{2}) + 6 \cdot - 9}{{(x^{2} - 9)}^{3}}$

Step 2.2.18.2

Simplify each term.

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Step 2.2.18.2.1

Multiply $- 3$ by $6$ .

$f'' (x) = - \frac{- 18 x^{2} + 6 \cdot - 9}{{(x^{2} - 9)}^{3}}$

Step 2.2.18.2.2

Multiply $6$ by $- 9$ .

$f'' (x) = - \frac{- 18 x^{2} - 54}{{(x^{2} - 9)}^{3}}$

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $- \frac{- 18 x^{2} - 54}{{(x^{2} - 9)}^{3}}$ .

$- \frac{- 18 x^{2} - 54}{{(x^{2} - 9)}^{3}}$

Step 3

Set the second derivative equal to $0$ then solve the equation $- \frac{- 18 x^{2} - 54}{{(x^{2} - 9)}^{3}} = 0$ .

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Step 3.1

Set the second derivative equal to $0$ .

$- \frac{- 18 x^{2} - 54}{{(x^{2} - 9)}^{3}} = 0$

Step 3.2

Set the numerator equal to zero.

$- 18 x^{2} - 54 = 0$

Step 3.3

Solve the equation for $x$ .

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Step 3.3.1

Add $54$ to both sides of the equation.

$- 18 x^{2} = 54$

Step 3.3.2

Divide each term in $- 18 x^{2} = 54$ by $- 18$ and simplify.

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Step 3.3.2.1

Divide each term in $- 18 x^{2} = 54$ by $- 18$ .

$\frac{- 18 x^{2}}{- 18} = \frac{54}{- 18}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Cancel the common factor of $- 18$ .

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Step 3.3.2.2.1.1

Cancel the common factor.

$\frac{- 18 x^{2}}{- 18} = \frac{54}{- 18}$

Step 3.3.2.2.1.2

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

$x^{2} = \frac{54}{- 18}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

Divide $54$ by $- 18$ .

$x^{2} = - 3$

Step 3.3.3

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

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

Step 3.3.4

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

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Step 3.3.4.1

Rewrite $- 3$ as $- 1 (3)$ .

$x = \pm \sqrt{- 1 (3)}$

Step 3.3.4.2

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

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

Step 3.3.4.3

Rewrite $\sqrt{- 1}$ as $i$ .

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

Step 3.3.5

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

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Step 3.3.5.1

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

$x = i \sqrt{3}$

Step 3.3.5.2

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

$x = - i \sqrt{3}$

Step 3.3.5.3

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

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

Step 4

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

No Inflection Points