Calculus Examples

$\frac{6}{x^{2} - 16}$

Step 1

Write $\frac{6}{x^{2} - 16}$ as a function.

$f (x) = \frac{6}{x^{2} - 16}$

Step 2

Find the first 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 $6$ is constant with respect to $x$ , the derivative of $\frac{6}{x^{2} - 16}$ with respect to $x$ is $6 \frac{d}{d x} [\frac{1}{x^{2} - 16}]$ .

$f' (x) = 6 \frac{d}{d x} (\frac{1}{x^{2} - 16})$

Step 2.1.1.2

Rewrite $\frac{1}{x^{2} - 16}$ as ${(x^{2} - 16)}^{- 1}$ .

$f' (x) = 6 \frac{d}{d x} ({(x^{2} - 16)}^{- 1})$

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 $g (x) = x^{2} - 16$ .

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Step 2.1.2.1

To apply the Chain Rule, set $u$ as $x^{2} - 16$ .

$f' (x) = 6 (\frac{d}{d u} (u^{- 1}) \frac{d}{d x} (x^{2} - 16))$

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 $n = - 1$ .

$f' (x) = 6 (- u^{- 2} \frac{d}{d x} (x^{2} - 16))$

Step 2.1.2.3

Replace all occurrences of $u$ with $x^{2} - 16$ .

$f' (x) = 6 (- {(x^{2} - 16)}^{- 2} \frac{d}{d x} (x^{2} - 16))$

Step 2.1.3

Differentiate.

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Step 2.1.3.1

Multiply $- 1$ by $6$ .

$f' (x) = - 6 ({(x^{2} - 16)}^{- 2} \frac{d}{d x} (x^{2} - 16))$

Step 2.1.3.2

By the Sum Rule, the derivative of $x^{2} - 16$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 16]$ .

$f' (x) = - 6 {(x^{2} - 16)}^{- 2} (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 16))$

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 $n = 2$ .

$f' (x) = - 6 {(x^{2} - 16)}^{- 2} (2 x + \frac{d}{d x} (- 16))$

Step 2.1.3.4

Since $- 16$ is constant with respect to $x$ , the derivative of $- 16$ with respect to $x$ is $0$ .

$f' (x) = - 6 {(x^{2} - 16)}^{- 2} (2 x + 0)$

Step 2.1.3.5

Simplify the expression.

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Step 2.1.3.5.1

Add $2 x$ and $0$ .

$f' (x) = - 6 {(x^{2} - 16)}^{- 2} (2 x)$

Step 2.1.3.5.2

Multiply $2$ by $- 6$ .

$f' (x) = - 12 {(x^{2} - 16)}^{- 2} x$

Step 2.1.4

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

$f' (x) = - 12 \frac{1}{{(x^{2} - 16)}^{2}} \cdot x$

Step 2.1.5

Combine terms.

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Step 2.1.5.1

Combine $- 12$ and $\frac{1}{{(x^{2} - 16)}^{2}}$ .

$f' (x) = \frac{- 12}{{(x^{2} - 16)}^{2}} \cdot x$

Step 2.1.5.2

Move the negative in front of the fraction.

$f' (x) = - \frac{12}{{(x^{2} - 16)}^{2}} \cdot x$

Step 2.1.5.3

Combine $x$ and $\frac{12}{{(x^{2} - 16)}^{2}}$ .

$f' (x) = - \frac{x \cdot 12}{{(x^{2} - 16)}^{2}}$

Step 2.1.5.4

Move $12$ to the left of $x$ .

$f' (x) = - \frac{12 x}{{(x^{2} - 16)}^{2}}$

Step 2.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{12 x}{{(x^{2} - 16)}^{2}}$ .

$- \frac{12 x}{{(x^{2} - 16)}^{2}}$

Step 3

Set the first derivative equal to $0$ then solve the equation $- \frac{12 x}{{(x^{2} - 16)}^{2}} = 0$ .

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Step 3.1

Set the first derivative equal to $0$ .

$- \frac{12 x}{{(x^{2} - 16)}^{2}} = 0$

Step 3.2

Set the numerator equal to zero.

$12 x = 0$

Step 3.3

Divide each term in $12 x = 0$ by $12$ and simplify.

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Step 3.3.1

Divide each term in $12 x = 0$ by $12$ .

$\frac{12 x}{12} = \frac{0}{12}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $12$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{12 x}{12} = \frac{0}{12}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{12}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $0$ by $12$ .

$x = 0$

Step 4

The values which make the derivative equal to $0$ are $0$ .

$0$

Step 5

Find where the derivative is undefined.

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Step 5.1

Set the denominator in $\frac{12 x}{{(x^{2} - 16)}^{2}}$ equal to $0$ to find where the expression is undefined.

${(x^{2} - 16)}^{2} = 0$

Step 5.2

Solve for $x$ .

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Step 5.2.1

Factor the left side of the equation.

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Step 5.2.1.1

Rewrite $16$ as $4^{2}$ .

${(x^{2} - 4^{2})}^{2} = 0$

Step 5.2.1.2

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 $b = 4$ .

${((x + 4) (x - 4))}^{2} = 0$

Step 5.2.1.3

Apply the product rule to $(x + 4) (x - 4)$ .

${(x + 4)}^{2} {(x - 4)}^{2} = 0$

Step 5.2.2

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

${(x + 4)}^{2} = 0$

${(x - 4)}^{2} = 0$

Step 5.2.3

Set ${(x + 4)}^{2}$ equal to $0$ and solve for $x$ .

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Step 5.2.3.1

Set ${(x + 4)}^{2}$ equal to $0$ .

${(x + 4)}^{2} = 0$

Step 5.2.3.2

Solve ${(x + 4)}^{2} = 0$ for $x$ .

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Step 5.2.3.2.1

Set the $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 5.2.3.2.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 5.2.4

Set ${(x - 4)}^{2}$ equal to $0$ and solve for $x$ .

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Step 5.2.4.1

Set ${(x - 4)}^{2}$ equal to $0$ .

${(x - 4)}^{2} = 0$

Step 5.2.4.2

Solve ${(x - 4)}^{2} = 0$ for $x$ .

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Step 5.2.4.2.1

Set the $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 5.2.4.2.2

Add $4$ to both sides of the equation.

$x = 4$

Step 5.2.5

The final solution is all the values that make ${(x + 4)}^{2} {(x - 4)}^{2} = 0$ true.

$x = - 4, 4$

Step 5.3

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 $0$ .

$x = - 4, x = 4$

Step 6

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the derivative $0$ or undefined.

$(- \infty, - 4) \cup (- 4, 0) \cup (0, 4) \cup (4, \infty)$

Step 7

Substitute a value from the interval $(- \infty, - 4)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $- 5$ in the expression.

$f' (- 5) = - \frac{12 (- 5)}{{({(- 5)}^{2} - 16)}^{2}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Multiply $12$ by $- 5$ .

$f' (- 5) = - \frac{- 60}{{({(- 5)}^{2} - 16)}^{2}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

Raise $- 5$ to the power of $2$ .

$f' (- 5) = - \frac{- 60}{{(25 - 16)}^{2}}$

Step 7.2.2.2

Subtract $16$ from $25$ .

$f' (- 5) = - \frac{- 60}{9^{2}}$

Step 7.2.2.3

Raise $9$ to the power of $2$ .

$f' (- 5) = - \frac{- 60}{81}$

Step 7.2.3

Reduce the expression by cancelling the common factors.

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Step 7.2.3.1

Cancel the common factor of $- 60$ and $81$ .

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Step 7.2.3.1.1

Factor $3$ out of $- 60$ .

$f' (- 5) = - \frac{3 (- 20)}{81}$

Step 7.2.3.1.2

Cancel the common factors.

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Step 7.2.3.1.2.1

Factor $3$ out of $81$ .

$f' (- 5) = - \frac{3 \cdot - 20}{3 \cdot 27}$

Step 7.2.3.1.2.2

Cancel the common factor.

$f' (- 5) = - \frac{3 \cdot - 20}{3 \cdot 27}$

Step 7.2.3.1.2.3

Rewrite the expression.

$f' (- 5) = - \frac{- 20}{27}$

Step 7.2.3.2

Move the negative in front of the fraction.

$f' (- 5) = \frac{20}{27}$

Step 7.2.4

The final answer is $\frac{20}{27}$ .

$\frac{20}{27}$

Step 7.3

At $x = - 5$ the derivative is $\frac{20}{27}$ . Since this is positive, the function is increasing on $(- \infty, - 4)$ .

Increasing on $(- \infty, - 4)$ since $f' (x) > 0$

Step 8

Substitute a value from the interval $(- 4, 0)$ into the derivative to determine if the function is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $- 2$ in the expression.

$f' (- 2) = - \frac{12 (- 2)}{{({(- 2)}^{2} - 16)}^{2}}$

Step 8.2

Simplify the result.

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Step 8.2.1

Multiply $12$ by $- 2$ .

$f' (- 2) = - \frac{- 24}{{({(- 2)}^{2} - 16)}^{2}}$

Step 8.2.2

Simplify the denominator.

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Step 8.2.2.1

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

$f' (- 2) = - \frac{- 24}{{(4 - 16)}^{2}}$

Step 8.2.2.2

Subtract $16$ from $4$ .

$f' (- 2) = - \frac{- 24}{{(- 12)}^{2}}$

Step 8.2.2.3

Raise $- 12$ to the power of $2$ .

$f' (- 2) = - \frac{- 24}{144}$

Step 8.2.3

Reduce the expression by cancelling the common factors.

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Step 8.2.3.1

Cancel the common factor of $- 24$ and $144$ .

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Step 8.2.3.1.1

Factor $24$ out of $- 24$ .

$f' (- 2) = - \frac{24 (- 1)}{144}$

Step 8.2.3.1.2

Cancel the common factors.

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Step 8.2.3.1.2.1

Factor $24$ out of $144$ .

$f' (- 2) = - \frac{24 \cdot - 1}{24 \cdot 6}$

Step 8.2.3.1.2.2

Cancel the common factor.

$f' (- 2) = - \frac{24 \cdot - 1}{24 \cdot 6}$

Step 8.2.3.1.2.3

Rewrite the expression.

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

Step 8.2.3.2

Move the negative in front of the fraction.

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

Step 8.2.4

The final answer is $\frac{1}{6}$ .

$\frac{1}{6}$

Step 8.3

At $x = - 2$ the derivative is $\frac{1}{6}$ . Since this is positive, the function is increasing on $(- 4, 0)$ .

Increasing on $(- 4, 0)$ since $f' (x) > 0$

Step 9

Substitute a value from the interval $(0, 4)$ into the derivative to determine if the function is increasing or decreasing.

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Step 9.1

Replace the variable $x$ with $2$ in the expression.

$f' (2) = - \frac{12 (2)}{{({(2)}^{2} - 16)}^{2}}$

Step 9.2

Simplify the result.

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Step 9.2.1

Multiply $12$ by $2$ .

$f' (2) = - \frac{24}{{(2^{2} - 16)}^{2}}$

Step 9.2.2

Simplify the denominator.

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Step 9.2.2.1

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

$f' (2) = - \frac{24}{{(4 - 16)}^{2}}$

Step 9.2.2.2

Subtract $16$ from $4$ .

$f' (2) = - \frac{24}{{(- 12)}^{2}}$

Step 9.2.2.3

Raise $- 12$ to the power of $2$ .

$f' (2) = - \frac{24}{144}$

Step 9.2.3

Cancel the common factor of $24$ and $144$ .

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Step 9.2.3.1

Factor $24$ out of $24$ .

$f' (2) = - \frac{24 (1)}{144}$

Step 9.2.3.2

Cancel the common factors.

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Step 9.2.3.2.1

Factor $24$ out of $144$ .

$f' (2) = - \frac{24 \cdot 1}{24 \cdot 6}$

Step 9.2.3.2.2

Cancel the common factor.

$f' (2) = - \frac{24 \cdot 1}{24 \cdot 6}$

Step 9.2.3.2.3

Rewrite the expression.

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

Step 9.2.4

The final answer is $- \frac{1}{6}$ .

$- \frac{1}{6}$

Step 9.3

At $x = 2$ the derivative is $- \frac{1}{6}$ . Since this is negative, the function is decreasing on $(0, 4)$ .

Decreasing on $(0, 4)$ since $f' (x) < 0$

Step 10

Substitute a value from the interval $(4, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 10.1

Replace the variable $x$ with $5$ in the expression.

$f' (5) = - \frac{12 (5)}{{({(5)}^{2} - 16)}^{2}}$

Step 10.2

Simplify the result.

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Step 10.2.1

Multiply $12$ by $5$ .

$f' (5) = - \frac{60}{{(5^{2} - 16)}^{2}}$

Step 10.2.2

Simplify the denominator.

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Step 10.2.2.1

Raise $5$ to the power of $2$ .

$f' (5) = - \frac{60}{{(25 - 16)}^{2}}$

Step 10.2.2.2

Subtract $16$ from $25$ .

$f' (5) = - \frac{60}{9^{2}}$

Step 10.2.2.3

Raise $9$ to the power of $2$ .

$f' (5) = - \frac{60}{81}$

Step 10.2.3

Cancel the common factor of $60$ and $81$ .

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Step 10.2.3.1

Factor $3$ out of $60$ .

$f' (5) = - \frac{3 (20)}{81}$

Step 10.2.3.2

Cancel the common factors.

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Step 10.2.3.2.1

Factor $3$ out of $81$ .

$f' (5) = - \frac{3 \cdot 20}{3 \cdot 27}$

Step 10.2.3.2.2

Cancel the common factor.

$f' (5) = - \frac{3 \cdot 20}{3 \cdot 27}$

Step 10.2.3.2.3

Rewrite the expression.

$f' (5) = - \frac{20}{27}$

Step 10.2.4

The final answer is $- \frac{20}{27}$ .

$- \frac{20}{27}$

Step 10.3

At $x = 5$ the derivative is $- \frac{20}{27}$ . Since this is negative, the function is decreasing on $(4, \infty)$ .

Decreasing on $(4, \infty)$ since $f' (x) < 0$

Step 11

List the intervals on which the function is increasing and decreasing.

Increasing on: $(- \infty, - 4), (- 4, 0)$

Decreasing on: $(0, 4), (4, \infty)$

Step 12