Calculus Examples

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

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

By the Sum Rule, the derivative of $x^{\frac{1}{9}} + 9$ with respect to $x$ is $\frac{d}{d x} [x^{\frac{1}{9}}] + \frac{d}{d x} [9]$ .

$\frac{d}{d x} [x^{\frac{1}{9}}] + \frac{d}{d x} [9]$

Step 1.1.2

Evaluate $\frac{d}{d x} [x^{\frac{1}{9}}]$ .

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Step 1.1.2.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{9}$ .

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

Step 1.1.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{1}{9} x^{\frac{1}{9} - 1 \cdot \frac{9}{9}} + \frac{d}{d x} [9]$

Step 1.1.2.3

Combine $- 1$ and $\frac{9}{9}$ .

$\frac{1}{9} x^{\frac{1}{9} + \frac{- 1 \cdot 9}{9}} + \frac{d}{d x} [9]$

Step 1.1.2.4

Combine the numerators over the common denominator.

$\frac{1}{9} x^{\frac{1 - 1 \cdot 9}{9}} + \frac{d}{d x} [9]$

Step 1.1.2.5

Simplify the numerator.

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Step 1.1.2.5.1

Multiply $- 1$ by $9$ .

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

Step 1.1.2.5.2

Subtract $9$ from $1$ .

$\frac{1}{9} x^{\frac{- 8}{9}} + \frac{d}{d x} [9]$

Step 1.1.2.6

Move the negative in front of the fraction.

$\frac{1}{9} x^{- \frac{8}{9}} + \frac{d}{d x} [9]$

Step 1.1.3

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

$\frac{1}{9} x^{- \frac{8}{9}} + 0$

Step 1.1.4

Simplify.

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Step 1.1.4.1

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

$\frac{1}{9} \cdot \frac{1}{x^{\frac{8}{9}}} + 0$

Step 1.1.4.2

Combine terms.

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Step 1.1.4.2.1

Multiply $\frac{1}{9}$ by $\frac{1}{x^{\frac{8}{9}}}$ .

$\frac{1}{9 x^{\frac{8}{9}}} + 0$

Step 1.1.4.2.2

Add $\frac{1}{9 x^{\frac{8}{9}}}$ and $0$ .

$f' (x) = \frac{1}{9 x^{\frac{8}{9}}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{1}{9 x^{\frac{8}{9}}}$ .

$\frac{1}{9 x^{\frac{8}{9}}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{1}{9 x^{\frac{8}{9}}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{1}{9 x^{\frac{8}{9}}} = 0$

Step 2.2

Set the numerator equal to zero.

$1 = 0$

Step 2.3

Since $1 \neq 0$ , there are no solutions.

No solution

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

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\frac{1}{9 \sqrt[9]{x^{8}}}$

Step 4.2

Set the denominator in $\frac{1}{9 \sqrt[9]{x^{8}}}$ equal to $0$ to find where the expression is undefined.

$9 \sqrt[9]{x^{8}} = 0$

Step 4.3

Solve for $x$ .

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Step 4.3.1

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $9$ .

${(9 \sqrt[9]{x^{8}})}^{9} = 0^{9}$

Step 4.3.2

Simplify each side of the equation.

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Step 4.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[9]{x^{8}}$ as $x^{\frac{8}{9}}$ .

${(9 x^{\frac{8}{9}})}^{9} = 0^{9}$

Step 4.3.2.2

Simplify the left side.

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Step 4.3.2.2.1

Simplify ${(9 x^{\frac{8}{9}})}^{9}$ .

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Step 4.3.2.2.1.1

Apply the product rule to $9 x^{\frac{8}{9}}$ .

$9^{9} {(x^{\frac{8}{9}})}^{9} = 0^{9}$

Step 4.3.2.2.1.2

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

$387420489 {(x^{\frac{8}{9}})}^{9} = 0^{9}$

Step 4.3.2.2.1.3

Multiply the exponents in ${(x^{\frac{8}{9}})}^{9}$ .

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Step 4.3.2.2.1.3.1

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

$387420489 x^{\frac{8}{9} \cdot 9} = 0^{9}$

Step 4.3.2.2.1.3.2

Cancel the common factor of $9$ .

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Step 4.3.2.2.1.3.2.1

Cancel the common factor.

$387420489 x^{\frac{8}{9} \cdot 9} = 0^{9}$

Step 4.3.2.2.1.3.2.2

Rewrite the expression.

$387420489 x^{8} = 0^{9}$

Step 4.3.2.3

Simplify the right side.

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Step 4.3.2.3.1

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

$387420489 x^{8} = 0$

Step 4.3.3

Solve for $x$ .

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Step 4.3.3.1

Divide each term in $387420489 x^{8} = 0$ by $387420489$ and simplify.

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Step 4.3.3.1.1

Divide each term in $387420489 x^{8} = 0$ by $387420489$ .

$\frac{387420489 x^{8}}{387420489} = \frac{0}{387420489}$

Step 4.3.3.1.2

Simplify the left side.

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Step 4.3.3.1.2.1

Cancel the common factor of $387420489$ .

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Step 4.3.3.1.2.1.1

Cancel the common factor.

$\frac{387420489 x^{8}}{387420489} = \frac{0}{387420489}$

Step 4.3.3.1.2.1.2

Divide $x^{8}$ by $1$ .

$x^{8} = \frac{0}{387420489}$

Step 4.3.3.1.3

Simplify the right side.

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Step 4.3.3.1.3.1

Divide $0$ by $387420489$ .

$x^{8} = 0$

Step 4.3.3.2

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

$x = \pm \sqrt[8]{0}$

Step 4.3.3.3

Simplify $\pm \sqrt[8]{0}$ .

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Step 4.3.3.3.1

Rewrite $0$ as $0^{8}$ .

$x = \pm \sqrt[8]{0^{8}}$

Step 4.3.3.3.2

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

$x = \pm 0$

Step 4.3.3.3.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 5

After finding the point that makes the derivative $f' (x) = \frac{1}{9 x^{\frac{8}{9}}}$ equal to $0$ or undefined, the interval to check where $f (x) = x^{\frac{1}{9}} + 9$ is increasing and where it is decreasing is $(- \infty, 0) \cup (0, \infty)$ .

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

Step 6

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

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Step 6.1

Replace the variable $x$ with $- 1$ in the expression.

$f' (- 1) = \frac{1}{9 {(- 1)}^{\frac{8}{9}}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the denominator.

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Step 6.2.1.1

Rewrite $- 1$ as ${(- 1)}^{9}$ .

$f' (- 1) = \frac{1}{9 {({(- 1)}^{9})}^{\frac{8}{9}}}$

Step 6.2.1.2

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

$f' (- 1) = \frac{1}{9 {(- 1)}^{9 (\frac{8}{9})}}$

Step 6.2.1.3

Cancel the common factor of $9$ .

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Step 6.2.1.3.1

Cancel the common factor.

$f' (- 1) = \frac{1}{9 {(- 1)}^{9 (\frac{8}{9})}}$

Step 6.2.1.3.2

Rewrite the expression.

$f' (- 1) = \frac{1}{9 {(- 1)}^{8}}$

Step 6.2.1.4

Raise $- 1$ to the power of $8$ .

$f' (- 1) = \frac{1}{9 \cdot 1}$

Step 6.2.2

Multiply $9$ by $1$ .

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

Step 6.2.3

The final answer is $\frac{1}{9}$ .

$\frac{1}{9}$

Step 6.3

At $x = - 1$ the derivative is $\frac{1}{9}$ . Since this is positive, the function is increasing on $(- \infty, 0)$ .

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

Step 7

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

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Step 7.1

Replace the variable $x$ with $1$ in the expression.

$f' (1) = \frac{1}{9 {(1)}^{\frac{8}{9}}}$

Step 7.2

Simplify the result.

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Step 7.2.1

One to any power is one.

$f' (1) = \frac{1}{9 \cdot 1}$

Step 7.2.2

Multiply $9$ by $1$ .

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

Step 7.2.3

The final answer is $\frac{1}{9}$ .

$\frac{1}{9}$

Step 7.3

At $x = 1$ the derivative is $\frac{1}{9}$ . Since this is positive, the function is increasing on $(0, \infty)$ .

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

Step 8

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

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

Step 9