Calculus Examples

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

Step 1

Find the first derivative.

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Find the first derivative.

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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Multiply $- 1$ by $3$ .

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

Subtract $3$ from $1$ .

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

Move the negative in front of the fraction.

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

Simplify.

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

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

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

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

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

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

Step 2

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

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Set the first derivative equal to $0$ .

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

Set the numerator equal to zero.

$1 = 0$

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

No solution

Step 3

Find the values where the derivative is undefined.

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

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

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

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

Solve for $x$ .

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

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

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

$27 {(x^{\frac{2}{3}})}^{3} = 0^{3}$

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

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

$27 x^{\frac{2}{3} \cdot 3} = 0^{3}$

Cancel the common factor of $3$ .

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Cancel the common factor.

$27 x^{\frac{2}{3} \cdot 3} = 0^{3}$

Rewrite the expression.

$27 x^{2} = 0^{3}$

Simplify the right side.

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

$27 x^{2} = 0$

Solve for $x$ .

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Divide each term in $27 x^{2} = 0$ by $27$ and simplify.

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Divide each term in $27 x^{2} = 0$ by $27$ .

$\frac{27 x^{2}}{27} = \frac{0}{27}$

Simplify the left side.

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Cancel the common factor of $27$ .

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Cancel the common factor.

$\frac{27 x^{2}}{27} = \frac{0}{27}$

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{0}{27}$

Simplify the right side.

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Divide $0$ by $27$ .

$x^{2} = 0$

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

$x = \pm \sqrt{0}$

Simplify $\pm \sqrt{0}$ .

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

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

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

$x = \pm 0$

Plus or minus $0$ is $0$ .

$x = 0$

Step 4

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

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Evaluate at $x = 0$ .

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Substitute $0$ for $x$ .

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

Simplify.

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Simplify the expression.

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

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

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

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

Cancel the common factor of $3$ .

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Cancel the common factor.

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

Rewrite the expression.

$0^{1}$

Evaluate the exponent.

$0$

List all of the points.

$(0, 0)$

Step 5