Calculus Examples

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

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

By the Sum Rule, the derivative of $x^{\frac{2}{3}} - x$ with respect to $x$ is $\frac{d}{d x} [x^{\frac{2}{3}}] + \frac{d}{d x} [- x]$ .

$\frac{d}{d x} [x^{\frac{2}{3}}] + \frac{d}{d x} [- x]$

Step 1.1.2

Evaluate $\frac{d}{d x} [x^{\frac{2}{3}}]$ .

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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 $n = \frac{2}{3}$ .

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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

Step 1.1.2.4

Combine the numerators over the common denominator.

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

Step 1.1.2.5

Simplify the numerator.

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Step 1.1.2.5.1

Multiply $- 1$ by $3$ .

$\frac{2}{3} x^{\frac{2 - 3}{3}} + \frac{d}{d x} [- x]$

Step 1.1.2.5.2

Subtract $3$ from $2$ .

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

Step 1.1.2.6

Move the negative in front of the fraction.

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

Multiply $- 1$ by $1$ .

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

Step 1.1.4

Simplify.

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Step 1.1.4.1

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

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

Step 1.1.4.2

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

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Add $1$ to both sides of the equation.

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

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.

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

Step 2.3.2

The LCM of one and any expression is the expression.

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

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Rewrite using the commutative property of multiplication.

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

Step 2.4.2.2

Cancel the common factor of $3$ .

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Step 2.4.2.2.1

Cancel the common factor.

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

Step 2.4.2.2.2

Rewrite the expression.

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

Step 2.4.2.3

Cancel the common factor of $x^{\frac{1}{3}}$ .

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Step 2.4.2.3.1

Cancel the common factor.

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

Step 2.4.2.3.2

Rewrite the expression.

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

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

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

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

Step 2.5

Solve the equation.

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Step 2.5.1

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

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

Step 2.5.2

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

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Step 2.5.2.1

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

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

Step 2.5.2.2

Simplify the left side.

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Step 2.5.2.2.1

Cancel the common factor.

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

Step 2.5.2.2.2

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

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

Step 2.5.3

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

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

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

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Step 2.5.4.1.1.1

Multiply the exponents in ${(x^{\frac{1}{3}})}^{3}$ .

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Step 2.5.4.1.1.1.1

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

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

Step 2.5.4.1.1.1.2

Cancel the common factor of $3$ .

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Step 2.5.4.1.1.1.2.1

Cancel the common factor.

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

Step 2.5.4.1.1.1.2.2

Rewrite the expression.

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

Step 2.5.4.1.1.2

Simplify.

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

Step 2.5.4.2

Simplify the right side.

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Step 2.5.4.2.1

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

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Step 2.5.4.2.1.1

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

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

Step 2.5.4.2.1.2

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

$x = \frac{8}{3^{3}}$

Step 2.5.4.2.1.3

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

$x = \frac{8}{27}$

Step 3

The values which make the derivative equal to $0$ are $\frac{8}{27}$ .

$\frac{8}{27}$

Step 4

Find where the derivative is undefined.

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Step 4.1

Convert expressions with fractional exponents to radicals.

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Step 4.1.1

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

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

Step 4.1.2

Anything raised to $1$ is the base itself.

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

Step 4.2

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

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

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, cube both sides of the equation.

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

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 $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

${(3 x^{\frac{1}{3}})}^{3} = 0^{3}$

Step 4.3.2.2

Simplify the left side.

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Step 4.3.2.2.1

Simplify ${(3 x^{\frac{1}{3}})}^{3}$ .

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Step 4.3.2.2.1.1

Apply the product rule to $3 x^{\frac{1}{3}}$ .

$3^{3} {(x^{\frac{1}{3}})}^{3} = 0^{3}$

Step 4.3.2.2.1.2

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

$27 {(x^{\frac{1}{3}})}^{3} = 0^{3}$

Step 4.3.2.2.1.3

Multiply the exponents in ${(x^{\frac{1}{3}})}^{3}$ .

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Step 4.3.2.2.1.3.1

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

$27 x^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 4.3.2.2.1.3.2

Cancel the common factor of $3$ .

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Step 4.3.2.2.1.3.2.1

Cancel the common factor.

$27 x^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 4.3.2.2.1.3.2.2

Rewrite the expression.

$27 x^{1} = 0^{3}$

Step 4.3.2.2.1.4

Simplify.

$27 x = 0^{3}$

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 $0$ .

$27 x = 0$

Step 4.3.3

Divide each term in $27 x = 0$ by $27$ and simplify.

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Step 4.3.3.1

Divide each term in $27 x = 0$ by $27$ .

$\frac{27 x}{27} = \frac{0}{27}$

Step 4.3.3.2

Simplify the left side.

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Step 4.3.3.2.1

Cancel the common factor of $27$ .

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Step 4.3.3.2.1.1

Cancel the common factor.

$\frac{27 x}{27} = \frac{0}{27}$

Step 4.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{27}$

Step 4.3.3.3

Simplify the right side.

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Step 4.3.3.3.1

Divide $0$ by $27$ .

$x = 0$

Step 5

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

$(- \infty, 0) \cup (0, \frac{8}{27}) \cup (\frac{8}{27}, \infty)$

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.

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

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Simplify the denominator.

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Step 6.2.1.1.1

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

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

Step 6.2.1.1.2

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

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

Step 6.2.1.1.3

Cancel the common factor of $3$ .

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Step 6.2.1.1.3.1

Cancel the common factor.

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

Step 6.2.1.1.3.2

Rewrite the expression.

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

Step 6.2.1.1.4

Evaluate the exponent.

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

Step 6.2.1.2

Multiply $3$ by $- 1$ .

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

Step 6.2.1.3

Move the negative in front of the fraction.

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

Step 6.2.2

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

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

Step 6.2.3

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

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

Step 6.2.4

Combine the numerators over the common denominator.

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

Step 6.2.5

Simplify the numerator.

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Step 6.2.5.1

Multiply $- 1$ by $3$ .

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

Step 6.2.5.2

Subtract $3$ from $- 2$ .

$f' (- 1) = \frac{- 5}{3}$

Step 6.2.6

Move the negative in front of the fraction.

$f' (- 1) = - \frac{5}{3}$

Step 6.2.7

The final answer is $- \frac{5}{3}$ .

$- \frac{5}{3}$

Step 6.3

At $x = - 1$ the derivative is $- \frac{5}{3}$ . Since this is negative, the function is decreasing on $(- \infty, 0)$ .

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

Step 7

Substitute a value from the interval $(0, \frac{8}{27})$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $0.\overline{148}$ in the expression.

$f' (0.\overline{148}) = \frac{2}{3 {(0.\overline{148})}^{\frac{1}{3}}} - 1$

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

Raise $0.\overline{148}$ to the power of $\frac{1}{3}$ .

$f' (0.\overline{148}) = \frac{2}{3 \cdot 0.52913368} - 1$

Step 7.2.1.2

Multiply $3$ by $0.52913368$ .

$f' (0.\overline{148}) = \frac{2}{1.58740105} - 1$

Step 7.2.1.3

Divide $2$ by $1.58740105$ .

$f' (0.\overline{148}) = 1.25992104 - 1$

Step 7.2.2

Subtract $1$ from $1.25992104$ .

$f' (0.\overline{148}) = 0.25992104$

Step 7.2.3

The final answer is $0.25992104$ .

$0.25992104$

Step 7.3

At $x = 0.\overline{148}$ the derivative is $0.25992104$ . Since this is positive, the function is increasing on $(0, \frac{8}{27})$ .

Increasing on $(0, \frac{8}{27})$ since $f' (x) > 0$

Step 8

Substitute a value from the interval $(\frac{8}{27}, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $1.2962963$ in the expression.

$f' (1.2962963) = \frac{2}{3 {(1.2962963)}^{\frac{1}{3}}} - 1$

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

Raise $1.2962963$ to the power of $\frac{1}{3}$ .

$f' (1.2962963) = \frac{2}{3 \cdot 1.09035543} - 1$

Step 8.2.1.2

Multiply $3$ by $1.09035543$ .

$f' (1.2962963) = \frac{2}{3.27106631} - 1$

Step 8.2.1.3

Divide $2$ by $3.27106631$ .

$f' (1.2962963) = 0.61142141 - 1$

Step 8.2.2

Subtract $1$ from $0.61142141$ .

$f' (1.2962963) = - 0.38857858$

Step 8.2.3

The final answer is $- 0.38857858$ .

$- 0.38857858$

Step 8.3

At $x = 1.2962963$ the derivative is $- 0.38857858$ . Since this is negative, the function is decreasing on $(\frac{8}{27}, \infty)$ .

Decreasing on $(\frac{8}{27}, \infty)$ since $f' (x) < 0$

Step 9

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

Increasing on: $(0, \frac{8}{27})$

Decreasing on: $(- \infty, 0), (\frac{8}{27}, \infty)$

Step 10