Calculus Examples

$f (x) = \sqrt[3]{x^{2}} + 2 x$

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 $\sqrt[3]{x^{2}} + 2 x$ with respect to $x$ is $\frac{d}{d x} [\sqrt[3]{x^{2}}] + \frac{d}{d x} [2 x]$ .

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

Step 1.1.2

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

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Step 1.1.2.1

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

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

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

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

Step 1.1.2.3

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

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

Step 1.1.2.4

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

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

Step 1.1.2.5

Combine the numerators over the common denominator.

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

Step 1.1.2.6

Simplify the numerator.

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Step 1.1.2.6.1

Multiply $- 1$ by $3$ .

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

Step 1.1.2.6.2

Subtract $3$ from $2$ .

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

Step 1.1.2.7

Move the negative in front of the fraction.

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

Step 1.1.3

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

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Step 1.1.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

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 $n = 1$ .

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

Step 1.1.3.3

Multiply $2$ by $1$ .

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

Step 1.1.4

Simplify.

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Step 1.1.4.1

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

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

Step 1.1.4.2

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

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Subtract $2$ from both sides of the equation.

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

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

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Step 2.4.1

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

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

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Rewrite using the commutative property of multiplication.

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

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.

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

Step 2.4.2.2.2

Rewrite the expression.

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

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.

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

Step 2.4.2.3.2

Rewrite the expression.

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

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Multiply $3$ by $- 2$ .

$2 = - 6 x^{\frac{1}{3}}$

Step 2.5

Solve the equation.

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Step 2.5.1

Rewrite the equation as $- 6 x^{\frac{1}{3}} = 2$ .

$- 6 x^{\frac{1}{3}} = 2$

Step 2.5.2

Divide each term in $- 6 x^{\frac{1}{3}} = 2$ by $- 6$ and simplify.

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Step 2.5.2.1

Divide each term in $- 6 x^{\frac{1}{3}} = 2$ by $- 6$ .

$\frac{- 6 x^{\frac{1}{3}}}{- 6} = \frac{2}{- 6}$

Step 2.5.2.2

Simplify the left side.

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Step 2.5.2.2.1

Cancel the common factor.

$\frac{- 6 x^{\frac{1}{3}}}{- 6} = \frac{2}{- 6}$

Step 2.5.2.2.2

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

$x^{\frac{1}{3}} = \frac{2}{- 6}$

Step 2.5.2.3

Simplify the right side.

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Step 2.5.2.3.1

Cancel the common factor of $2$ and $- 6$ .

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Step 2.5.2.3.1.1

Factor $2$ out of $2$ .

$x^{\frac{1}{3}} = \frac{2 (1)}{- 6}$

Step 2.5.2.3.1.2

Cancel the common factors.

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Step 2.5.2.3.1.2.1

Factor $2$ out of $- 6$ .

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

Step 2.5.2.3.1.2.2

Cancel the common factor.

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

Step 2.5.2.3.1.2.3

Rewrite the expression.

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

Step 2.5.2.3.2

Move the negative in front of the fraction.

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

Step 2.5.3

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

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

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.

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

Step 2.5.4.1.1.1.2.2

Rewrite the expression.

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

Step 2.5.4.1.1.2

Simplify.

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

Step 2.5.4.2

Simplify the right side.

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Step 2.5.4.2.1

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

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Step 2.5.4.2.1.1

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

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Step 2.5.4.2.1.1.1

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

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

Step 2.5.4.2.1.1.2

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

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

Step 2.5.4.2.1.2

Raise $- 1$ to the power of $3$ .

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

Step 2.5.4.2.1.3

One to any power is one.

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

Step 2.5.4.2.1.4

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

$x = - \frac{1}{27}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Convert expressions with fractional exponents to radicals.

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Step 3.1.1

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

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

Step 3.1.2

Anything raised to $1$ is the base itself.

$\frac{2}{3 \sqrt[3]{x}} + 2$

Step 3.2

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 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.

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

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

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

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

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

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Step 3.3.2.2.1.1

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

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

Step 3.3.2.2.1.2

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

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

Step 3.3.2.2.1.3

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

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Step 3.3.2.2.1.3.1

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

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

Step 3.3.2.2.1.3.2

Cancel the common factor of $3$ .

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Step 3.3.2.2.1.3.2.1

Cancel the common factor.

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

Step 3.3.2.2.1.3.2.2

Rewrite the expression.

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

Step 3.3.2.2.1.4

Simplify.

$27 x = 0^{3}$

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 $0$ .

$27 x = 0$

Step 3.3.3

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

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Step 3.3.3.1

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

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

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $27$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

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

Step 3.3.3.2.1.2

Divide $x$ by $1$ .

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

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Divide $0$ by $27$ .

$x = 0$

Step 4

Evaluate $\sqrt[3]{x^{2}} + 2 x$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = - \frac{1}{27}$ .

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Step 4.1.1

Substitute $- \frac{1}{27}$ for $x$ .

$\sqrt[3]{{(- \frac{1}{27})}^{2}} + 2 (- \frac{1}{27})$

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

Apply the product rule to $- \frac{1}{27}$ .

$\sqrt[3]{{(- 1)}^{2} {(\frac{1}{27})}^{2}} + 2 (- \frac{1}{27})$

Step 4.1.2.1.2

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

$\sqrt[3]{1 {(\frac{1}{27})}^{2}} + 2 (- \frac{1}{27})$

Step 4.1.2.1.3

Apply the product rule to $\frac{1}{27}$ .

$\sqrt[3]{1 \frac{1^{2}}{27^{2}}} + 2 (- \frac{1}{27})$

Step 4.1.2.1.4

One to any power is one.

$\sqrt[3]{1 \frac{1}{27^{2}}} + 2 (- \frac{1}{27})$

Step 4.1.2.1.5

Raise $27$ to the power of $2$ .

$\sqrt[3]{1 (\frac{1}{729})} + 2 (- \frac{1}{27})$

Step 4.1.2.1.6

Multiply $\frac{1}{729}$ by $1$ .

$\sqrt[3]{\frac{1}{729}} + 2 (- \frac{1}{27})$

Step 4.1.2.1.7

Rewrite $\sqrt[3]{\frac{1}{729}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{729}}$ .

$\frac{\sqrt[3]{1}}{\sqrt[3]{729}} + 2 (- \frac{1}{27})$

Step 4.1.2.1.8

Any root of $1$ is $1$ .

$\frac{1}{\sqrt[3]{729}} + 2 (- \frac{1}{27})$

Step 4.1.2.1.9

Simplify the denominator.

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Step 4.1.2.1.9.1

Rewrite $729$ as $9^{3}$ .

$\frac{1}{\sqrt[3]{9^{3}}} + 2 (- \frac{1}{27})$

Step 4.1.2.1.9.2

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

$\frac{1}{9} + 2 (- \frac{1}{27})$

Step 4.1.2.1.10

Multiply $2 (- \frac{1}{27})$ .

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Step 4.1.2.1.10.1

Multiply $- 1$ by $2$ .

$\frac{1}{9} - 2 (\frac{1}{27})$

Step 4.1.2.1.10.2

Combine $- 2$ and $\frac{1}{27}$ .

$\frac{1}{9} + \frac{- 2}{27}$

Step 4.1.2.1.11

Move the negative in front of the fraction.

$\frac{1}{9} - \frac{2}{27}$

Step 4.1.2.2

To write $\frac{1}{9}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{9} \cdot \frac{3}{3} - \frac{2}{27}$

Step 4.1.2.3

Write each expression with a common denominator of $27$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.2.3.1

Multiply $\frac{1}{9}$ by $\frac{3}{3}$ .

$\frac{3}{9 \cdot 3} - \frac{2}{27}$

Step 4.1.2.3.2

Multiply $9$ by $3$ .

$\frac{3}{27} - \frac{2}{27}$

Step 4.1.2.4

Combine the numerators over the common denominator.

$\frac{3 - 2}{27}$

Step 4.1.2.5

Subtract $2$ from $3$ .

$\frac{1}{27}$

Step 4.2

Evaluate at $x = 0$ .

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Step 4.2.1

Substitute $0$ for $x$ .

$\sqrt[3]{{(0)}^{2}} + 2 (0)$

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

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

$\sqrt[3]{0} + 2 (0)$

Step 4.2.2.1.2

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

$\sqrt[3]{0^{3}} + 2 (0)$

Step 4.2.2.1.3

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

$0 + 2 (0)$

Step 4.2.2.1.4

Multiply $2$ by $0$ .

$0 + 0$

Step 4.2.2.2

Add $0$ and $0$ .

$0$

Step 4.3

List all of the points.

$(- \frac{1}{27}, \frac{1}{27}), (0, 0)$

Step 5