Calculus Examples

$f (x) = \frac{2 x}{x^{2} - 36}$

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 $2$ is constant with respect to $x$ , the derivative of $\frac{2 x}{x^{2} - 36}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{x}{x^{2} - 36}]$ .

$2 \frac{d}{d x} [\frac{x}{x^{2} - 36}]$

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 $g (x) = x^{2} - 36$ .

$2 \frac{(x^{2} - 36) \frac{d}{d x} [x] - x \frac{d}{d x} [x^{2} - 36]}{{(x^{2} - 36)}^{2}}$

Step 1.1.3

Differentiate.

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Step 1.1.3.1

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

$2 \frac{(x^{2} - 36) \cdot 1 - x \frac{d}{d x} [x^{2} - 36]}{{(x^{2} - 36)}^{2}}$

Step 1.1.3.2

Multiply $x^{2} - 36$ by $1$ .

$2 \frac{x^{2} - 36 - x \frac{d}{d x} [x^{2} - 36]}{{(x^{2} - 36)}^{2}}$

Step 1.1.3.3

By the Sum Rule, the derivative of $x^{2} - 36$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 36]$ .

$2 \frac{x^{2} - 36 - x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 36])}{{(x^{2} - 36)}^{2}}$

Step 1.1.3.4

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

$2 \frac{x^{2} - 36 - x (2 x + \frac{d}{d x} [- 36])}{{(x^{2} - 36)}^{2}}$

Step 1.1.3.5

Since $- 36$ is constant with respect to $x$ , the derivative of $- 36$ with respect to $x$ is $0$ .

$2 \frac{x^{2} - 36 - x (2 x + 0)}{{(x^{2} - 36)}^{2}}$

Step 1.1.3.6

Simplify the expression.

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Step 1.1.3.6.1

Add $2 x$ and $0$ .

$2 \frac{x^{2} - 36 - x (2 x)}{{(x^{2} - 36)}^{2}}$

Step 1.1.3.6.2

Multiply $2$ by $- 1$ .

$2 \frac{x^{2} - 36 - 2 x \cdot x}{{(x^{2} - 36)}^{2}}$

Step 1.1.4

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

$2 \frac{x^{2} - 36 - 2 (x^{1} x)}{{(x^{2} - 36)}^{2}}$

Step 1.1.5

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

$2 \frac{x^{2} - 36 - 2 (x^{1} x^{1})}{{(x^{2} - 36)}^{2}}$

Step 1.1.6

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

$2 \frac{x^{2} - 36 - 2 x^{1 + 1}}{{(x^{2} - 36)}^{2}}$

Step 1.1.7

Add $1$ and $1$ .

$2 \frac{x^{2} - 36 - 2 x^{2}}{{(x^{2} - 36)}^{2}}$

Step 1.1.8

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

$2 \frac{- x^{2} - 36}{{(x^{2} - 36)}^{2}}$

Step 1.1.9

Combine $2$ and $\frac{- x^{2} - 36}{{(x^{2} - 36)}^{2}}$ .

$\frac{2 (- x^{2} - 36)}{{(x^{2} - 36)}^{2}}$

Step 1.1.10

Simplify.

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Step 1.1.10.1

Apply the distributive property.

$\frac{2 (- x^{2}) + 2 \cdot - 36}{{(x^{2} - 36)}^{2}}$

Step 1.1.10.2

Simplify each term.

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Step 1.1.10.2.1

Multiply $- 1$ by $2$ .

$\frac{- 2 x^{2} + 2 \cdot - 36}{{(x^{2} - 36)}^{2}}$

Step 1.1.10.2.2

Multiply $2$ by $- 36$ .

$f' (x) = \frac{- 2 x^{2} - 72}{{(x^{2} - 36)}^{2}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{- 2 x^{2} - 72}{{(x^{2} - 36)}^{2}}$ .

$\frac{- 2 x^{2} - 72}{{(x^{2} - 36)}^{2}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{- 2 x^{2} - 72}{{(x^{2} - 36)}^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{- 2 x^{2} - 72}{{(x^{2} - 36)}^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

$- 2 x^{2} - 72 = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Add $72$ to both sides of the equation.

$- 2 x^{2} = 72$

Step 2.3.2

Divide each term in $- 2 x^{2} = 72$ by $- 2$ and simplify.

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Step 2.3.2.1

Divide each term in $- 2 x^{2} = 72$ by $- 2$ .

$\frac{- 2 x^{2}}{- 2} = \frac{72}{- 2}$

Step 2.3.2.2

Simplify the left side.

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Step 2.3.2.2.1

Cancel the common factor of $- 2$ .

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Step 2.3.2.2.1.1

Cancel the common factor.

$\frac{- 2 x^{2}}{- 2} = \frac{72}{- 2}$

Step 2.3.2.2.1.2

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

$x^{2} = \frac{72}{- 2}$

Step 2.3.2.3

Simplify the right side.

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Step 2.3.2.3.1

Divide $72$ by $- 2$ .

$x^{2} = - 36$

Step 2.3.3

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

$x = \pm \sqrt{- 36}$

Step 2.3.4

Simplify $\pm \sqrt{- 36}$ .

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Step 2.3.4.1

Rewrite $- 36$ as $- 1 (36)$ .

$x = \pm \sqrt{- 1 (36)}$

Step 2.3.4.2

Rewrite $\sqrt{- 1 (36)}$ as $\sqrt{- 1} \cdot \sqrt{36}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{36}$

Step 2.3.4.3

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

$x = \pm i \cdot \sqrt{36}$

Step 2.3.4.4

Rewrite $36$ as $6^{2}$ .

$x = \pm i \cdot \sqrt{6^{2}}$

Step 2.3.4.5

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm i \cdot 6$

Step 2.3.4.6

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

$x = \pm 6 i$

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.

$x = 6 i$

Step 2.3.5.2

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

$x = - 6 i$

Step 2.3.5.3

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

$x = 6 i, - 6 i$

Step 3

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

No critical points found

Step 4

Find where the derivative is undefined.

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Step 4.1

Set the denominator in $\frac{- 2 x^{2} - 72}{{(x^{2} - 36)}^{2}}$ equal to $0$ to find where the expression is undefined.

${(x^{2} - 36)}^{2} = 0$

Step 4.2

Solve for $x$ .

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Step 4.2.1

Factor the left side of the equation.

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Step 4.2.1.1

Rewrite $36$ as $6^{2}$ .

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

Step 4.2.1.2

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 $b = 6$ .

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

Step 4.2.1.3

Apply the product rule to $(x + 6) (x - 6)$ .

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

Step 4.2.2

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

${(x + 6)}^{2} = 0$

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

Step 4.2.3

Set ${(x + 6)}^{2}$ equal to $0$ and solve for $x$ .

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Step 4.2.3.1

Set ${(x + 6)}^{2}$ equal to $0$ .

${(x + 6)}^{2} = 0$

Step 4.2.3.2

Solve ${(x + 6)}^{2} = 0$ for $x$ .

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Step 4.2.3.2.1

Set the $x + 6$ equal to $0$ .

$x + 6 = 0$

Step 4.2.3.2.2

Subtract $6$ from both sides of the equation.

$x = - 6$

Step 4.2.4

Set ${(x - 6)}^{2}$ equal to $0$ and solve for $x$ .

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Step 4.2.4.1

Set ${(x - 6)}^{2}$ equal to $0$ .

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

Step 4.2.4.2

Solve ${(x - 6)}^{2} = 0$ for $x$ .

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Step 4.2.4.2.1

Set the $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 4.2.4.2.2

Add $6$ to both sides of the equation.

$x = 6$

Step 4.2.5

The final solution is all the values that make ${(x + 6)}^{2} {(x - 6)}^{2} = 0$ true.

$x = - 6, 6$

Step 4.3

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 $0$ .

$x = - 6, x = 6$

Step 5

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

$(- \infty, - 6) \cup (- 6, 6) \cup (6, \infty)$

Step 6

Substitute a value from the interval $(- \infty, - 6)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 7$ in the expression.

$f' (- 7) = \frac{- 2 {(- 7)}^{2} - 72}{{({(- 7)}^{2} - 36)}^{2}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the numerator.

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Step 6.2.1.1

Raise $- 7$ to the power of $2$ .

$f' (- 7) = \frac{- 2 \cdot 49 - 72}{{({(- 7)}^{2} - 36)}^{2}}$

Step 6.2.1.2

Multiply $- 2$ by $49$ .

$f' (- 7) = \frac{- 98 - 72}{{({(- 7)}^{2} - 36)}^{2}}$

Step 6.2.1.3

Subtract $72$ from $- 98$ .

$f' (- 7) = \frac{- 170}{{({(- 7)}^{2} - 36)}^{2}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Raise $- 7$ to the power of $2$ .

$f' (- 7) = \frac{- 170}{{(49 - 36)}^{2}}$

Step 6.2.2.2

Subtract $36$ from $49$ .

$f' (- 7) = \frac{- 170}{13^{2}}$

Step 6.2.2.3

Raise $13$ to the power of $2$ .

$f' (- 7) = \frac{- 170}{169}$

Step 6.2.3

Move the negative in front of the fraction.

$f' (- 7) = - \frac{170}{169}$

Step 6.2.4

The final answer is $- \frac{170}{169}$ .

$- \frac{170}{169}$

Step 6.3

At $x = - 7$ the derivative is $- \frac{170}{169}$ . Since this is negative, the function is decreasing on $(- \infty, - 6)$ .

Decreasing on $(- \infty, - 6)$ since $f' (x) < 0$

Step 7

Substitute a value from the interval $(- 6, 6)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $0$ in the expression.

$f' (0) = \frac{- 2 {(0)}^{2} - 72}{{({(0)}^{2} - 36)}^{2}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify the numerator.

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Step 7.2.1.1

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

$f' (0) = \frac{- 2 \cdot 0 - 72}{{(0^{2} - 36)}^{2}}$

Step 7.2.1.2

Multiply $- 2$ by $0$ .

$f' (0) = \frac{0 - 72}{{(0^{2} - 36)}^{2}}$

Step 7.2.1.3

Subtract $72$ from $0$ .

$f' (0) = \frac{- 72}{{(0^{2} - 36)}^{2}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

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

$f' (0) = \frac{- 72}{{(0 - 36)}^{2}}$

Step 7.2.2.2

Subtract $36$ from $0$ .

$f' (0) = \frac{- 72}{{(- 36)}^{2}}$

Step 7.2.2.3

Raise $- 36$ to the power of $2$ .

$f' (0) = \frac{- 72}{1296}$

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 $- 72$ and $1296$ .

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Step 7.2.3.1.1

Factor $72$ out of $- 72$ .

$f' (0) = \frac{72 (- 1)}{1296}$

Step 7.2.3.1.2

Cancel the common factors.

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Step 7.2.3.1.2.1

Factor $72$ out of $1296$ .

$f' (0) = \frac{72 \cdot - 1}{72 \cdot 18}$

Step 7.2.3.1.2.2

Cancel the common factor.

$f' (0) = \frac{72 \cdot - 1}{72 \cdot 18}$

Step 7.2.3.1.2.3

Rewrite the expression.

$f' (0) = \frac{- 1}{18}$

Step 7.2.3.2

Move the negative in front of the fraction.

$f' (0) = - \frac{1}{18}$

Step 7.2.4

The final answer is $- \frac{1}{18}$ .

$- \frac{1}{18}$

Step 7.3

At $x = 0$ the derivative is $- \frac{1}{18}$ . Since this is negative, the function is decreasing on $(- 6, 6)$ .

Decreasing on $(- 6, 6)$ since $f' (x) < 0$

Step 8

Substitute a value from the interval $(6, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $7$ in the expression.

$f' (7) = \frac{- 2 {(7)}^{2} - 72}{{({(7)}^{2} - 36)}^{2}}$

Step 8.2

Simplify the result.

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Step 8.2.1

Simplify the numerator.

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Step 8.2.1.1

Raise $7$ to the power of $2$ .

$f' (7) = \frac{- 2 \cdot 49 - 72}{{(7^{2} - 36)}^{2}}$

Step 8.2.1.2

Multiply $- 2$ by $49$ .

$f' (7) = \frac{- 98 - 72}{{(7^{2} - 36)}^{2}}$

Step 8.2.1.3

Subtract $72$ from $- 98$ .

$f' (7) = \frac{- 170}{{(7^{2} - 36)}^{2}}$

Step 8.2.2

Simplify the denominator.

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Step 8.2.2.1

Raise $7$ to the power of $2$ .

$f' (7) = \frac{- 170}{{(49 - 36)}^{2}}$

Step 8.2.2.2

Subtract $36$ from $49$ .

$f' (7) = \frac{- 170}{13^{2}}$

Step 8.2.2.3

Raise $13$ to the power of $2$ .

$f' (7) = \frac{- 170}{169}$

Step 8.2.3

Move the negative in front of the fraction.

$f' (7) = - \frac{170}{169}$

Step 8.2.4

The final answer is $- \frac{170}{169}$ .

$- \frac{170}{169}$

Step 8.3

At $x = 7$ the derivative is $- \frac{170}{169}$ . Since this is negative, the function is decreasing on $(6, \infty)$ .

Decreasing on $(6, \infty)$ since $f' (x) < 0$

Step 9

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

Decreasing on: $(- \infty, - 6), (- 6, 6), (6, \infty)$

Step 10