Calculus Examples

$f (x) = x + \frac{9}{x}$

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

Differentiate.

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Step 1.1.1.1

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

$f' (x) = \frac{d}{d x} (x) + \frac{d}{d x} (\frac{9}{x})$

Step 1.1.1.2

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

$f' (x) = 1 + \frac{d}{d x} (\frac{9}{x})$

Step 1.1.2

Evaluate $\frac{d}{d x} [\frac{9}{x}]$ .

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Step 1.1.2.1

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

$f' (x) = 1 + 9 \frac{d}{d x} (\frac{1}{x})$

Step 1.1.2.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

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

Step 1.1.2.3

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

$f' (x) = 1 + 9 (- x^{- 2})$

Step 1.1.2.4

Multiply $- 1$ by $9$ .

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

Step 1.1.3

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

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

Step 1.1.4

Simplify.

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Step 1.1.4.1

Combine terms.

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Step 1.1.4.1.1

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

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

Step 1.1.4.1.2

Move the negative in front of the fraction.

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

Step 1.1.4.2

Reorder terms.

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

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{9}{x^{2}} + 1$ .

$- \frac{9}{x^{2}} + 1$

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{9}{x^{2}} + 1 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- \frac{9}{x^{2}} + 1 = 0$

Step 2.2

Subtract $1$ from both sides of the equation.

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

Step 2.3

Find the LCD of the terms in the equation.

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Step 2.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x^{2}, 1$

Step 2.3.2

The LCM of one and any expression is the expression.

$x^{2}$

Step 2.4

Multiply each term in $- \frac{9}{x^{2}} = - 1$ by $x^{2}$ to eliminate the fractions.

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Step 2.4.1

Multiply each term in $- \frac{9}{x^{2}} = - 1$ by $x^{2}$ .

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

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $x^{2}$ .

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Step 2.4.2.1.1

Move the leading negative in $- \frac{9}{x^{2}}$ into the numerator.

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

Step 2.4.2.1.2

Cancel the common factor.

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

Step 2.4.2.1.3

Rewrite the expression.

$- 9 = - x^{2}$

Step 2.5

Solve the equation.

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Step 2.5.1

Rewrite the equation as $- x^{2} = - 9$ .

$- x^{2} = - 9$

Step 2.5.2

Divide each term in $- x^{2} = - 9$ by $- 1$ and simplify.

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Step 2.5.2.1

Divide each term in $- x^{2} = - 9$ by $- 1$ .

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

Step 2.5.2.2

Simplify the left side.

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Step 2.5.2.2.1

Dividing two negative values results in a positive value.

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

Step 2.5.2.2.2

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

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

Step 2.5.2.3

Simplify the right side.

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Step 2.5.2.3.1

Divide $- 9$ by $- 1$ .

$x^{2} = 9$

Step 2.5.3

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

$x = \pm \sqrt{9}$

Step 2.5.4

Simplify $\pm \sqrt{9}$ .

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Step 2.5.4.1

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

$x = \pm \sqrt{3^{2}}$

Step 2.5.4.2

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

$x = \pm 3$

Step 2.5.5

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

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Step 2.5.5.1

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

$x = 3$

Step 2.5.5.2

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

$x = - 3$

Step 2.5.5.3

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

$x = 3, - 3$

Step 3

The values which make the derivative equal to $0$ are $3, - 3$ .

$3, - 3$

Step 4

Find where the derivative is undefined.

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Step 4.1

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

$x^{2} = 0$

Step 4.2

Solve for $x$ .

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Step 4.2.1

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

$x = \pm \sqrt{0}$

Step 4.2.2

Simplify $\pm \sqrt{0}$ .

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Step 4.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 4.2.2.2

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

$x = \pm 0$

Step 4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 5

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

$(- \infty, - 3) \cup (- 3, 0) \cup (0, 3) \cup (3, \infty)$

Step 6

Substitute a value from the interval $(- \infty, - 3)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 4$ in the expression.

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

Step 6.2

Simplify the result.

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Step 6.2.1

Raise $- 4$ to the power of $2$ .

$f' (- 4) = - \frac{9}{16} + 1$

Step 6.2.2

Write $1$ as a fraction with a common denominator.

$f' (- 4) = - \frac{9}{16} + \frac{16}{16}$

Step 6.2.3

Combine the numerators over the common denominator.

$f' (- 4) = \frac{- 9 + 16}{16}$

Step 6.2.4

Add $- 9$ and $16$ .

$f' (- 4) = \frac{7}{16}$

Step 6.2.5

The final answer is $\frac{7}{16}$ .

$\frac{7}{16}$

Step 6.3

At $x = - 4$ the derivative is $\frac{7}{16}$ . Since this is positive, the function is increasing on $(- \infty, - 3)$ .

Increasing on $(- \infty, - 3)$ since $f' (x) > 0$

Step 7

Substitute a value from the interval $(- 3, 0)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $- \frac{3}{2}$ in the expression.

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

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Simplify the denominator.

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Step 7.2.1.1.1

Apply the product rule to $- \frac{3}{2}$ .

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

Step 7.2.1.1.2

Raise $- 1$ to the power of $2$ .

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

Step 7.2.1.1.3

Apply the product rule to $\frac{3}{2}$ .

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

Step 7.2.1.1.4

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

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

Step 7.2.1.1.5

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

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

Step 7.2.1.1.6

Multiply $\frac{9}{4}$ by $1$ .

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

Step 7.2.1.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 7.2.1.3

Cancel the common factor of $9$ .

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Step 7.2.1.3.1

Cancel the common factor.

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

Step 7.2.1.3.2

Rewrite the expression.

$f' (- \frac{3}{2}) = - 1 \cdot 4 + 1$

Step 7.2.1.4

Multiply $- 1$ by $4$ .

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

Step 7.2.2

Add $- 4$ and $1$ .

$f' (- \frac{3}{2}) = - 3$

Step 7.2.3

The final answer is $- 3$ .

$- 3$

Step 7.3

At $x = - \frac{3}{2}$ the derivative is $- 3$ . Since this is negative, the function is decreasing on $(- 3, 0)$ .

Decreasing on $(- 3, 0)$ since $f' (x) < 0$

Step 8

Substitute a value from the interval $(0, 3)$ into the derivative to determine if the function is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $\frac{3}{2}$ in the expression.

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

Step 8.2

Simplify the result.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

Simplify the denominator.

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Step 8.2.1.1.1

Apply the product rule to $\frac{3}{2}$ .

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

Step 8.2.1.1.2

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

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

Step 8.2.1.1.3

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

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

Step 8.2.1.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.2.1.3

Cancel the common factor of $9$ .

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Step 8.2.1.3.1

Cancel the common factor.

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

Step 8.2.1.3.2

Rewrite the expression.

$f' (\frac{3}{2}) = - 1 \cdot 4 + 1$

Step 8.2.1.4

Multiply $- 1$ by $4$ .

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

Step 8.2.2

Add $- 4$ and $1$ .

$f' (\frac{3}{2}) = - 3$

Step 8.2.3

The final answer is $- 3$ .

$- 3$

Step 8.3

At $x = \frac{3}{2}$ the derivative is $- 3$ . Since this is negative, the function is decreasing on $(0, 3)$ .

Decreasing on $(0, 3)$ since $f' (x) < 0$

Step 9

Substitute a value from the interval $(3, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 9.1

Replace the variable $x$ with $4$ in the expression.

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

Step 9.2

Simplify the result.

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Step 9.2.1

Raise $4$ to the power of $2$ .

$f' (4) = - \frac{9}{16} + 1$

Step 9.2.2

Simplify the expression.

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Step 9.2.2.1

Write $1$ as a fraction with a common denominator.

$f' (4) = - \frac{9}{16} + \frac{16}{16}$

Step 9.2.2.2

Combine the numerators over the common denominator.

$f' (4) = \frac{- 9 + 16}{16}$

Step 9.2.2.3

Add $- 9$ and $16$ .

$f' (4) = \frac{7}{16}$

Step 9.2.3

The final answer is $\frac{7}{16}$ .

$\frac{7}{16}$

Step 9.3

At $x = 4$ the derivative is $\frac{7}{16}$ . Since this is positive, the function is increasing on $(3, \infty)$ .

Increasing on $(3, \infty)$ since $f' (x) > 0$

Step 10

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

Increasing on: $(- \infty, - 3), (3, \infty)$

Decreasing on: $(- 3, 0), (0, 3)$

Step 11