Calculus Examples

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

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

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

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

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

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Step 1.1.2.1

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

$\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2} - 9]$

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 $n = - 1$ .

$- u^{- 2} \frac{d}{d x} [x^{2} - 9]$

Step 1.1.2.3

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

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

Step 1.1.3

Differentiate.

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Step 1.1.3.1

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

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

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 $n = 2$ .

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

Step 1.1.3.3

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

$- {(x^{2} - 9)}^{- 2} (2 x + 0)$

Step 1.1.3.4

Simplify the expression.

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Step 1.1.3.4.1

Add $2 x$ and $0$ .

$- {(x^{2} - 9)}^{- 2} (2 x)$

Step 1.1.3.4.2

Multiply $2$ by $- 1$ .

$- 2 {(x^{2} - 9)}^{- 2} x$

Step 1.1.4

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

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

Step 1.1.5

Combine terms.

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Step 1.1.5.1

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

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

Step 1.1.5.2

Move the negative in front of the fraction.

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

Step 1.1.5.3

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

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

Step 1.1.5.4

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

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Set the numerator equal to zero.

$2 x = 0$

Step 2.3

Divide each term in $2 x = 0$ by $2$ and simplify.

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Step 2.3.1

Divide each term in $2 x = 0$ by $2$ .

$\frac{2 x}{2} = \frac{0}{2}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $2$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{0}{2}$

Step 2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{2}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $0$ by $2$ .

$x = 0$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

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

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

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 $3^{2}$ .

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

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 $b = 3$ .

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

Step 3.2.1.3

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

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

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 $0$ .

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

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

Step 3.2.3

Set ${(x + 3)}^{2}$ equal to $0$ and solve for $x$ .

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Step 3.2.3.1

Set ${(x + 3)}^{2}$ equal to $0$ .

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

Step 3.2.3.2

Solve ${(x + 3)}^{2} = 0$ 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 ${(x - 3)}^{2}$ equal to $0$ and solve for $x$ .

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Step 3.2.4.1

Set ${(x - 3)}^{2}$ equal to $0$ .

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

Step 3.2.4.2

Solve ${(x - 3)}^{2} = 0$ 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 ${(x + 3)}^{2} {(x - 3)}^{2} = 0$ 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 $\frac{1}{x^{2} - 9}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 0$ .

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Step 4.1.1

Substitute $0$ for $x$ .

$\frac{1}{{(0)}^{2} - 9}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify the denominator.

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Step 4.1.2.1.1

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

$\frac{1}{0 - 9}$

Step 4.1.2.1.2

Subtract $9$ from $0$ .

$\frac{1}{- 9}$

Step 4.1.2.2

Move the negative in front of the fraction.

$- \frac{1}{9}$

Step 4.2

Evaluate at $x = - 3$ .

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Step 4.2.1

Substitute $- 3$ for $x$ .

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

Step 4.2.2

Simplify.

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Step 4.2.2.1

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

$\frac{1}{9 - 9}$

Step 4.2.2.2

Subtract $9$ from $9$ .

$\frac{1}{0}$

Step 4.2.2.3

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

Undefined

Step 4.3

Evaluate at $x = 3$ .

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Step 4.3.1

Substitute $3$ for $x$ .

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

Step 4.3.2

Simplify.

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Step 4.3.2.1

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

$\frac{1}{9 - 9}$

Step 4.3.2.2

Subtract $9$ from $9$ .

$\frac{1}{0}$

Step 4.3.2.3

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

Undefined

Step 4.4

List all of the points.

$(0, - \frac{1}{9})$

Step 5