Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Since $- 18$ is constant with respect to $x$ , the derivative of $- \frac{18 x}{{(x^{2} - 9)}^{2}}$ with respect to $x$ is $- 18 \frac{d}{d x} [\frac{x}{{(x^{2} - 9)}^{2}}]$ .

$- 18 \frac{d}{d x} [\frac{x}{{(x^{2} - 9)}^{2}}]$

Step 1.1.2

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

$- 18 \frac{{(x^{2} - 9)}^{2} \frac{d}{d x} [x] - x \frac{d}{d x} [{(x^{2} - 9)}^{2}]}{{({(x^{2} - 9)}^{2})}^{2}}$

Step 1.1.3

Differentiate using the Power Rule.

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Step 1.1.3.1

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

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Step 1.1.3.1.1

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

$- 18 \frac{{(x^{2} - 9)}^{2} \frac{d}{d x} [x] - x \frac{d}{d x} [{(x^{2} - 9)}^{2}]}{{(x^{2} - 9)}^{2 \cdot 2}}$

Step 1.1.3.1.2

Multiply $2$ by $2$ .

$- 18 \frac{{(x^{2} - 9)}^{2} \frac{d}{d x} [x] - x \frac{d}{d x} [{(x^{2} - 9)}^{2}]}{{(x^{2} - 9)}^{4}}$

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 $n = 1$ .

$- 18 \frac{{(x^{2} - 9)}^{2} \cdot 1 - x \frac{d}{d x} [{(x^{2} - 9)}^{2}]}{{(x^{2} - 9)}^{4}}$

Step 1.1.3.3

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

$- 18 \frac{{(x^{2} - 9)}^{2} - x \frac{d}{d x} [{(x^{2} - 9)}^{2}]}{{(x^{2} - 9)}^{4}}$

Step 1.1.4

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 1.1.4.1

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

$- 18 \frac{{(x^{2} - 9)}^{2} - x (\frac{d}{d u} [u^{2}] \frac{d}{d x} [x^{2} - 9])}{{(x^{2} - 9)}^{4}}$

Step 1.1.4.2

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

$- 18 \frac{{(x^{2} - 9)}^{2} - x (2 u \frac{d}{d x} [x^{2} - 9])}{{(x^{2} - 9)}^{4}}$

Step 1.1.4.3

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

$- 18 \frac{{(x^{2} - 9)}^{2} - x (2 (x^{2} - 9) \frac{d}{d x} [x^{2} - 9])}{{(x^{2} - 9)}^{4}}$

Step 1.1.5

Simplify with factoring out.

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Step 1.1.5.1

Multiply $2$ by $- 1$ .

$- 18 \frac{{(x^{2} - 9)}^{2} - 2 x ((x^{2} - 9) \frac{d}{d x} [x^{2} - 9])}{{(x^{2} - 9)}^{4}}$

Step 1.1.5.2

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 1.1.5.2.1

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

$- 18 \frac{(x^{2} - 9) (x^{2} - 9) - 2 x ((x^{2} - 9) \frac{d}{d x} [x^{2} - 9])}{{(x^{2} - 9)}^{4}}$

Step 1.1.5.2.2

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

$- 18 \frac{(x^{2} - 9) (x^{2} - 9) + (x^{2} - 9) (- 2 x (\frac{d}{d x} [x^{2} - 9]))}{{(x^{2} - 9)}^{4}}$

Step 1.1.5.2.3

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

$- 18 \frac{(x^{2} - 9) (x^{2} - 9 - 2 x (\frac{d}{d x} [x^{2} - 9]))}{{(x^{2} - 9)}^{4}}$

Step 1.1.6

Cancel the common factors.

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Step 1.1.6.1

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

$- 18 \frac{(x^{2} - 9) (x^{2} - 9 - 2 x (\frac{d}{d x} [x^{2} - 9]))}{(x^{2} - 9) {(x^{2} - 9)}^{3}}$

Step 1.1.6.2

Cancel the common factor.

$- 18 \frac{(x^{2} - 9) (x^{2} - 9 - 2 x (\frac{d}{d x} [x^{2} - 9]))}{(x^{2} - 9) {(x^{2} - 9)}^{3}}$

Step 1.1.6.3

Rewrite the expression.

$- 18 \frac{x^{2} - 9 - 2 x (\frac{d}{d x} [x^{2} - 9])}{{(x^{2} - 9)}^{3}}$

Step 1.1.7

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

$- 18 \frac{x^{2} - 9 - 2 x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 9])}{{(x^{2} - 9)}^{3}}$

Step 1.1.8

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

$- 18 \frac{x^{2} - 9 - 2 x (2 x + \frac{d}{d x} [- 9])}{{(x^{2} - 9)}^{3}}$

Step 1.1.9

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

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

Step 1.1.10

Simplify the expression.

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Step 1.1.10.1

Add $2 x$ and $0$ .

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

Step 1.1.10.2

Multiply $2$ by $- 2$ .

$- 18 \frac{x^{2} - 9 - 4 x \cdot x}{{(x^{2} - 9)}^{3}}$

Step 1.1.11

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

$- 18 \frac{x^{2} - 9 - 4 (x^{1} x)}{{(x^{2} - 9)}^{3}}$

Step 1.1.12

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

$- 18 \frac{x^{2} - 9 - 4 (x^{1} x^{1})}{{(x^{2} - 9)}^{3}}$

Step 1.1.13

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

$- 18 \frac{x^{2} - 9 - 4 x^{1 + 1}}{{(x^{2} - 9)}^{3}}$

Step 1.1.14

Add $1$ and $1$ .

$- 18 \frac{x^{2} - 9 - 4 x^{2}}{{(x^{2} - 9)}^{3}}$

Step 1.1.15

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

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

Step 1.1.16

Combine $- 18$ and $\frac{- 3 x^{2} - 9}{{(x^{2} - 9)}^{3}}$ .

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

Step 1.1.17

Move the negative in front of the fraction.

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

Step 1.1.18

Simplify.

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Step 1.1.18.1

Apply the distributive property.

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

Step 1.1.18.2

Simplify each term.

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Step 1.1.18.2.1

Multiply $- 3$ by $18$ .

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

Step 1.1.18.2.2

Multiply $18$ by $- 9$ .

$f' (x) = - \frac{- 54 x^{2} - 162}{{(x^{2} - 9)}^{3}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{- 54 x^{2} - 162}{{(x^{2} - 9)}^{3}}$ .

$- \frac{- 54 x^{2} - 162}{{(x^{2} - 9)}^{3}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{- 54 x^{2} - 162}{{(x^{2} - 9)}^{3}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- \frac{- 54 x^{2} - 162}{{(x^{2} - 9)}^{3}} = 0$

Step 2.2

Set the numerator equal to zero.

$- 54 x^{2} - 162 = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Add $162$ to both sides of the equation.

$- 54 x^{2} = 162$

Step 2.3.2

Divide each term in $- 54 x^{2} = 162$ by $- 54$ and simplify.

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Step 2.3.2.1

Divide each term in $- 54 x^{2} = 162$ by $- 54$ .

$\frac{- 54 x^{2}}{- 54} = \frac{162}{- 54}$

Step 2.3.2.2

Simplify the left side.

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Step 2.3.2.2.1

Cancel the common factor of $- 54$ .

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Step 2.3.2.2.1.1

Cancel the common factor.

$\frac{- 54 x^{2}}{- 54} = \frac{162}{- 54}$

Step 2.3.2.2.1.2

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

$x^{2} = \frac{162}{- 54}$

Step 2.3.2.3

Simplify the right side.

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Step 2.3.2.3.1

Divide $162$ by $- 54$ .

$x^{2} = - 3$

Step 2.3.3

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

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

Step 2.3.4

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

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Step 2.3.4.1

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

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

Step 2.3.4.2

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

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

Step 2.3.4.3

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

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

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.

$x = i \sqrt{3}$

Step 2.3.5.2

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

$x = - i \sqrt{3}$

Step 2.3.5.3

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

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

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{- 54 x^{2} - 162}{{(x^{2} - 9)}^{3}}$ equal to $0$ to find where the expression is undefined.

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

Step 3.2

Solve for $x$ .

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Step 3.2.1

Factor the left side of the equation.

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Step 3.2.1.1

Rewrite $9$ as $3^{2}$ .

${(x^{2} - 3^{2})}^{3} = 0$

Step 3.2.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

${((x + 3) (x - 3))}^{3} = 0$

Step 3.2.1.3

Apply the product rule to $(x + 3) (x - 3)$ .

${(x + 3)}^{3} {(x - 3)}^{3} = 0$

Step 3.2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

${(x + 3)}^{3} = 0$

${(x - 3)}^{3} = 0$

Step 3.2.3

Set ${(x + 3)}^{3}$ equal to $0$ and solve for $x$ .

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Step 3.2.3.1

Set ${(x + 3)}^{3}$ equal to $0$ .

${(x + 3)}^{3} = 0$

Step 3.2.3.2

Solve ${(x + 3)}^{3} = 0$ for $x$ .

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Step 3.2.3.2.1

Set the $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 3.2.3.2.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 3.2.4

Set ${(x - 3)}^{3}$ equal to $0$ and solve for $x$ .

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Step 3.2.4.1

Set ${(x - 3)}^{3}$ equal to $0$ .

${(x - 3)}^{3} = 0$

Step 3.2.4.2

Solve ${(x - 3)}^{3} = 0$ for $x$ .

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Step 3.2.4.2.1

Set the $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 3.2.4.2.2

Add $3$ to both sides of the equation.

$x = 3$

Step 3.2.5

The final solution is all the values that make ${(x + 3)}^{3} {(x - 3)}^{3} = 0$ true.

$x = - 3, 3$

Step 3.3

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = - 3, x = 3$

Step 4

Evaluate $- \frac{18 x}{{(x^{2} - 9)}^{2}}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = - 3$ .

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Step 4.1.1

Substitute $- 3$ for $x$ .

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

Step 4.1.2

Simplify.

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Step 4.1.2.1

Raise $- 3$ to the power of $2$ .

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

Step 4.1.2.2

Subtract $9$ from $9$ .

$- \frac{18 (- 3)}{0^{2}}$

Step 4.1.2.3

Raising $0$ to any positive power yields $0$ .

$- \frac{18 (- 3)}{0}$

Step 4.1.2.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 4.2

Evaluate at $x = 3$ .

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Step 4.2.1

Substitute $3$ for $x$ .

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

Step 4.2.2

Simplify.

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Step 4.2.2.1

Raise $3$ to the power of $2$ .

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

Step 4.2.2.2

Subtract $9$ from $9$ .

$- \frac{18 (3)}{0^{2}}$

Step 4.2.2.3

Raising $0$ to any positive power yields $0$ .

$- \frac{18 (3)}{0}$

Step 4.2.2.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 5

There are no values of $x$ in the domain of the original problem where the derivative is $0$ or undefined.

No critical points found