Calculus Examples

$f (x) = x - 3 x^{\frac{1}{3}}$

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 $x - 3 x^{\frac{1}{3}}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 3 x^{\frac{1}{3}}]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 3 x^{\frac{1}{3}}]$

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 $n = 1$ .

$1 + \frac{d}{d x} [- 3 x^{\frac{1}{3}}]$

Step 1.1.2

Evaluate $\frac{d}{d x} [- 3 x^{\frac{1}{3}}]$ .

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Step 1.1.2.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{\frac{1}{3}}$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{\frac{1}{3}}]$ .

$1 - 3 \frac{d}{d x} [x^{\frac{1}{3}}]$

Step 1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{3}$ .

$1 - 3 (\frac{1}{3} x^{\frac{1}{3} - 1})$

Step 1.1.2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$1 - 3 (\frac{1}{3} x^{\frac{1}{3} - 1 \cdot \frac{3}{3}})$

Step 1.1.2.4

Combine $- 1$ and $\frac{3}{3}$ .

$1 - 3 (\frac{1}{3} x^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}})$

Step 1.1.2.5

Combine the numerators over the common denominator.

$1 - 3 (\frac{1}{3} x^{\frac{1 - 1 \cdot 3}{3}})$

Step 1.1.2.6

Simplify the numerator.

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Step 1.1.2.6.1

Multiply $- 1$ by $3$ .

$1 - 3 (\frac{1}{3} x^{\frac{1 - 3}{3}})$

Step 1.1.2.6.2

Subtract $3$ from $1$ .

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

Step 1.1.2.7

Move the negative in front of the fraction.

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

Step 1.1.2.8

Combine $\frac{1}{3}$ and $x^{- \frac{2}{3}}$ .

$1 - 3 \frac{x^{- \frac{2}{3}}}{3}$

Step 1.1.2.9

Combine $- 3$ and $\frac{x^{- \frac{2}{3}}}{3}$ .

$1 + \frac{- 3 x^{- \frac{2}{3}}}{3}$

Step 1.1.2.10

Move $x^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$1 + \frac{- 3}{3 x^{\frac{2}{3}}}$

Step 1.1.2.11

Factor $3$ out of $- 3$ .

$1 + \frac{3 \cdot - 1}{3 x^{\frac{2}{3}}}$

Step 1.1.2.12

Cancel the common factors.

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Step 1.1.2.12.1

Factor $3$ out of $3 x^{\frac{2}{3}}$ .

$1 + \frac{3 \cdot - 1}{3 (x^{\frac{2}{3}})}$

Step 1.1.2.12.2

Cancel the common factor.

$1 + \frac{3 \cdot - 1}{3 x^{\frac{2}{3}}}$

Step 1.1.2.12.3

Rewrite the expression.

$1 + \frac{- 1}{x^{\frac{2}{3}}}$

Step 1.1.2.13

Move the negative in front of the fraction.

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

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $1 - \frac{1}{x^{\frac{2}{3}}}$ .

$1 - \frac{1}{x^{\frac{2}{3}}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $1 - \frac{1}{x^{\frac{2}{3}}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$1 - \frac{1}{x^{\frac{2}{3}}} = 0$

Step 2.2

Subtract $1$ from both sides of the equation.

$- \frac{1}{x^{\frac{2}{3}}} = - 1$

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.

$x^{\frac{2}{3}}, 1$

Step 2.3.2

The LCM of one and any expression is the expression.

$x^{\frac{2}{3}}$

Step 2.4

Multiply each term in $- \frac{1}{x^{\frac{2}{3}}} = - 1$ by $x^{\frac{2}{3}}$ to eliminate the fractions.

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Step 2.4.1

Multiply each term in $- \frac{1}{x^{\frac{2}{3}}} = - 1$ by $x^{\frac{2}{3}}$ .

$- \frac{1}{x^{\frac{2}{3}}} x^{\frac{2}{3}} = - x^{\frac{2}{3}}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $x^{\frac{2}{3}}$ .

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Step 2.4.2.1.1

Move the leading negative in $- \frac{1}{x^{\frac{2}{3}}}$ into the numerator.

$\frac{- 1}{x^{\frac{2}{3}}} x^{\frac{2}{3}} = - x^{\frac{2}{3}}$

Step 2.4.2.1.2

Cancel the common factor.

$\frac{- 1}{x^{\frac{2}{3}}} x^{\frac{2}{3}} = - x^{\frac{2}{3}}$

Step 2.4.2.1.3

Rewrite the expression.

$- 1 = - x^{\frac{2}{3}}$

Step 2.5

Solve the equation.

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Step 2.5.1

Rewrite the equation as $- x^{\frac{2}{3}} = - 1$ .

$- x^{\frac{2}{3}} = - 1$

Step 2.5.2

Divide each term in $- x^{\frac{2}{3}} = - 1$ by $- 1$ and simplify.

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Step 2.5.2.1

Divide each term in $- x^{\frac{2}{3}} = - 1$ by $- 1$ .

$\frac{- x^{\frac{2}{3}}}{- 1} = \frac{- 1}{- 1}$

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.

$\frac{x^{\frac{2}{3}}}{1} = \frac{- 1}{- 1}$

Step 2.5.2.2.2

Divide $x^{\frac{2}{3}}$ by $1$ .

$x^{\frac{2}{3}} = \frac{- 1}{- 1}$

Step 2.5.2.3

Simplify the right side.

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Step 2.5.2.3.1

Divide $- 1$ by $- 1$ .

$x^{\frac{2}{3}} = 1$

Step 2.5.3

Raise each side of the equation to the power of $\frac{3}{2}$ to eliminate the fractional exponent on the left side.

${(x^{\frac{2}{3}})}^{\frac{3}{2}} = \pm 1^{\frac{3}{2}}$

Step 2.5.4

Simplify the exponent.

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Step 2.5.4.1

Simplify the left side.

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Step 2.5.4.1.1

Simplify ${(x^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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Step 2.5.4.1.1.1

Multiply the exponents in ${(x^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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Step 2.5.4.1.1.1.1

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

$x^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 1^{\frac{3}{2}}$

Step 2.5.4.1.1.1.2

Cancel the common factor of $2$ .

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Step 2.5.4.1.1.1.2.1

Cancel the common factor.

$x^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 1^{\frac{3}{2}}$

Step 2.5.4.1.1.1.2.2

Rewrite the expression.

$x^{\frac{1}{3} \cdot 3} = \pm 1^{\frac{3}{2}}$

Step 2.5.4.1.1.1.3

Cancel the common factor of $3$ .

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Step 2.5.4.1.1.1.3.1

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = \pm 1^{\frac{3}{2}}$

Step 2.5.4.1.1.1.3.2

Rewrite the expression.

$x^{1} = \pm 1^{\frac{3}{2}}$

Step 2.5.4.1.1.2

Simplify.

$x = \pm 1^{\frac{3}{2}}$

Step 2.5.4.2

Simplify the right side.

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Step 2.5.4.2.1

One to any power is one.

$x = \pm 1$

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.

$x = 1$

Step 2.5.5.2

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

$x = - 1$

Step 2.5.5.3

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

$x = 1, - 1$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

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

$1 - \frac{1}{\sqrt[3]{x^{2}}}$

Step 3.2

Set the denominator in $\frac{1}{\sqrt[3]{x^{2}}}$ equal to $0$ to find where the expression is undefined.

$\sqrt[3]{x^{2}} = 0$

Step 3.3

Solve for $x$ .

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Step 3.3.1

To remove the radical on the left side of the equation, cube both sides of the equation.

${\sqrt[3]{x^{2}}}^{3} = 0^{3}$

Step 3.3.2

Simplify each side of the equation.

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Step 3.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

${(x^{\frac{2}{3}})}^{3} = 0^{3}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Multiply the exponents in ${(x^{\frac{2}{3}})}^{3}$ .

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Step 3.3.2.2.1.1

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

$x^{\frac{2}{3} \cdot 3} = 0^{3}$

Step 3.3.2.2.1.2

Cancel the common factor of $3$ .

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Step 3.3.2.2.1.2.1

Cancel the common factor.

$x^{\frac{2}{3} \cdot 3} = 0^{3}$

Step 3.3.2.2.1.2.2

Rewrite the expression.

$x^{2} = 0^{3}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

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

$x^{2} = 0$

Step 3.3.3

Solve for $x$ .

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Step 3.3.3.1

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

$x = \pm \sqrt{0}$

Step 3.3.3.2

Simplify $\pm \sqrt{0}$ .

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Step 3.3.3.2.1

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

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

Step 3.3.3.2.2

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

$x = \pm 0$

Step 3.3.3.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 4

Evaluate $x - 3 x^{\frac{1}{3}}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 1$ .

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Step 4.1.1

Substitute $1$ for $x$ .

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

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

One to any power is one.

$1 - 3 \cdot 1$

Step 4.1.2.1.2

Multiply $- 3$ by $1$ .

$1 - 3$

Step 4.1.2.2

Subtract $3$ from $1$ .

$- 2$

Step 4.2

Evaluate at $x = - 1$ .

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Step 4.2.1

Substitute $- 1$ for $x$ .

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

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Rewrite $- 1$ as ${(- 1)}^{3}$ .

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

Step 4.2.2.1.2

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

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

Step 4.2.2.1.3

Cancel the common factor of $3$ .

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Step 4.2.2.1.3.1

Cancel the common factor.

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

Step 4.2.2.1.3.2

Rewrite the expression.

$- 1 - 3 {(- 1)}^{1}$

Step 4.2.2.1.4

Evaluate the exponent.

$- 1 - 3 \cdot - 1$

Step 4.2.2.1.5

Multiply $- 3$ by $- 1$ .

$- 1 + 3$

Step 4.2.2.2

Add $- 1$ and $3$ .

$2$

Step 4.3

Evaluate at $x = 0$ .

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Step 4.3.1

Substitute $0$ for $x$ .

$(0) - 3 {(0)}^{\frac{1}{3}}$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

Rewrite $0$ as $0^{3}$ .

$0 - 3 {(0^{3})}^{\frac{1}{3}}$

Step 4.3.2.1.2

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

$0 - 3 \cdot 0^{3 (\frac{1}{3})}$

Step 4.3.2.1.3

Cancel the common factor of $3$ .

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Step 4.3.2.1.3.1

Cancel the common factor.

$0 - 3 \cdot 0^{3 (\frac{1}{3})}$

Step 4.3.2.1.3.2

Rewrite the expression.

$0 - 3 \cdot 0^{1}$

Step 4.3.2.1.4

Evaluate the exponent.

$0 - 3 \cdot 0$

Step 4.3.2.1.5

Multiply $- 3$ by $0$ .

$0 + 0$

Step 4.3.2.2

Add $0$ and $0$ .

$0$

Step 4.4

List all of the points.

$(1, - 2), (- 1, 2), (0, 0)$

Step 5