Calculus Examples

$x - 3 \sqrt[3]{x}$

Step 1

Write $x - 3 \sqrt[3]{x}$ as a function.

$f (x) = x - 3 \sqrt[3]{x}$

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.

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Step 2.1.1.1

By the Sum Rule, the derivative of $x - 3 \sqrt[3]{x}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 3 \sqrt[3]{x}]$ .

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

Step 2.1.1.2

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

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

Step 2.1.2

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

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Step 2.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

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

Step 2.1.2.2

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 2.1.2.3

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 2.1.2.4

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 2.1.2.5

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

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

Step 2.1.2.6

Combine the numerators over the common denominator.

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

Step 2.1.2.7

Simplify the numerator.

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Step 2.1.2.7.1

Multiply $- 1$ by $3$ .

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

Step 2.1.2.7.2

Subtract $3$ from $1$ .

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

Step 2.1.2.8

Move the negative in front of the fraction.

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

Step 2.1.2.9

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

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

Step 2.1.2.10

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

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

Step 2.1.2.11

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 2.1.2.12

Factor $3$ out of $- 3$ .

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

Step 2.1.2.13

Cancel the common factors.

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Step 2.1.2.13.1

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

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

Step 2.1.2.13.2

Cancel the common factor.

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

Step 2.1.2.13.3

Rewrite the expression.

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

Step 2.1.2.14

Move the negative in front of the fraction.

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

Step 2.2

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 3

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

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Step 3.1

Set the first derivative equal to $0$ .

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

Step 3.2

Subtract $1$ from both sides of the equation.

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

Step 3.3

Find the LCD of the terms in the equation.

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Step 3.3.1

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 3.3.2

The LCM of one and any expression is the expression.

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

Step 3.4

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 3.4.1

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 3.4.2

Simplify the left side.

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Step 3.4.2.1

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

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Step 3.4.2.1.1

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 3.4.2.1.2

Cancel the common factor.

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

Step 3.4.2.1.3

Rewrite the expression.

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

Step 3.5

Solve the equation.

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Step 3.5.1

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

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

Step 3.5.2

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

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Step 3.5.2.1

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

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

Step 3.5.2.2

Simplify the left side.

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Step 3.5.2.2.1

Dividing two negative values results in a positive value.

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

Step 3.5.2.2.2

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

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

Step 3.5.2.3

Simplify the right side.

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Step 3.5.2.3.1

Divide $- 1$ by $- 1$ .

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

Step 3.5.3

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 3.5.4

Simplify the exponent.

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Step 3.5.4.1

Simplify the left side.

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Step 3.5.4.1.1

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

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Step 3.5.4.1.1.1

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

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Step 3.5.4.1.1.1.1

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 3.5.4.1.1.1.2

Cancel the common factor of $2$ .

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Step 3.5.4.1.1.1.2.1

Cancel the common factor.

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

Step 3.5.4.1.1.1.2.2

Rewrite the expression.

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

Step 3.5.4.1.1.1.3

Cancel the common factor of $3$ .

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Step 3.5.4.1.1.1.3.1

Cancel the common factor.

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

Step 3.5.4.1.1.1.3.2

Rewrite the expression.

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

Step 3.5.4.1.1.2

Simplify.

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

Step 3.5.4.2

Simplify the right side.

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Step 3.5.4.2.1

One to any power is one.

$x = \pm 1$

Step 3.5.5

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

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Step 3.5.5.1

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

$x = 1$

Step 3.5.5.2

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

$x = - 1$

Step 3.5.5.3

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

$x = 1, - 1$

Step 4

The values which make the derivative equal to $0$ are $1, - 1$ .

$1, - 1$

Step 5

Find where the derivative is undefined.

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Step 5.1

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 5.2

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 5.3

Solve for $x$ .

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Step 5.3.1

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 5.3.2

Simplify each side of the equation.

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Step 5.3.2.1

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 5.3.2.2

Simplify the left side.

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Step 5.3.2.2.1

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

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Step 5.3.2.2.1.1

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

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

Step 5.3.2.2.1.2

Cancel the common factor of $3$ .

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Step 5.3.2.2.1.2.1

Cancel the common factor.

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

Step 5.3.2.2.1.2.2

Rewrite the expression.

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

Step 5.3.2.3

Simplify the right side.

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Step 5.3.2.3.1

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

$x^{2} = 0$

Step 5.3.3

Solve for $x$ .

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Step 5.3.3.1

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

$x = \pm \sqrt{0}$

Step 5.3.3.2

Simplify $\pm \sqrt{0}$ .

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Step 5.3.3.2.1

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

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

Step 5.3.3.2.2

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

$x = \pm 0$

Step 5.3.3.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 6

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

$(- \infty, - 1) \cup (- 1, 0) \cup (0, 1) \cup (1, \infty)$

Step 7

Substitute a value from the interval $(- \infty, - 1)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

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

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

Step 7.2

The final answer is $1 - \frac{1}{{(- 2)}^{\frac{2}{3}}}$ .

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

Step 7.3

At $x = - 2$ the derivative is $1 - \frac{1}{{(- 2)}^{\frac{2}{3}}}$ . Since this is negative, the function is decreasing on $(- \infty, - 1)$ .

Decreasing on $(- \infty, - 1)$ since $f' (x) < 0$

Step 8

Substitute a value from the interval $(- 1, 0)$ into the derivative to determine if the function is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $- \frac{1}{2}$ in the expression.

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

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

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 8.2.1.1.1.1

Apply the product rule to $- \frac{1}{2}$ .

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

Step 8.2.1.1.1.2

Apply the product rule to $\frac{1}{2}$ .

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

Step 8.2.1.1.2

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

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

Step 8.2.1.1.3

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

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

Step 8.2.1.1.4

Cancel the common factor of $3$ .

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Step 8.2.1.1.4.1

Cancel the common factor.

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

Step 8.2.1.1.4.2

Rewrite the expression.

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

Step 8.2.1.1.5

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

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

Step 8.2.1.1.6

Multiply $\frac{1^{\frac{2}{3}}}{2^{\frac{2}{3}}}$ by $1$ .

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

Step 8.2.1.1.7

One to any power is one.

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

Step 8.2.1.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.2.1.3

Multiply $2^{\frac{2}{3}}$ by $1$ .

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

Step 8.2.2

The final answer is $1 - 2^{\frac{2}{3}}$ .

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

Step 8.3

Simplify.

$- 0.58740105$

Step 8.4

At $x = - \frac{1}{2}$ the derivative is $- 0.58740105$ . Since this is negative, the function is decreasing on $(- 1, 0)$ .

Decreasing on $(- 1, 0)$ since $f' (x) < 0$

Step 9

Substitute a value from the interval $(0, 1)$ into the derivative to determine if the function is increasing or decreasing.

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Step 9.1

Replace the variable $x$ with $\frac{1}{2}$ in the expression.

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

Step 9.2

Simplify the result.

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Step 9.2.1

Simplify each term.

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Step 9.2.1.1

Simplify the denominator.

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Step 9.2.1.1.1

Apply the product rule to $\frac{1}{2}$ .

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

Step 9.2.1.1.2

One to any power is one.

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

Step 9.2.1.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 9.2.1.3

Multiply $2^{\frac{2}{3}}$ by $1$ .

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

Step 9.2.2

The final answer is $1 - 2^{\frac{2}{3}}$ .

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

Step 9.3

Simplify.

$- 0.58740105$

Step 9.4

At $x = \frac{1}{2}$ the derivative is $- 0.58740105$ . Since this is negative, the function is decreasing on $(0, 1)$ .

Decreasing on $(0, 1)$ since $f' (x) < 0$

Step 10

Substitute a value from the interval $(1, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 10.1

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

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

Step 10.2

Simplify the result.

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Step 10.2.1

Remove parentheses.

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

Step 10.2.2

The final answer is $1 - \frac{1}{2^{\frac{2}{3}}}$ .

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

Step 10.3

Simplify.

$0.37003947$

Step 10.4

At $x = 2$ the derivative is $0.37003947$ . Since this is positive, the function is increasing on $(1, \infty)$ .

Increasing on $(1, \infty)$ since $f' (x) > 0$

Step 11

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

Increasing on: $(1, \infty)$

Decreasing on: $(- \infty, - 1), (- 1, 0), (0, 1)$

Step 12