Calculus Examples

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

Step 1

Write $x^{\frac{2}{3}}$ as a function.

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

Step 2

Find the first derivative of the function.

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Step 2.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{2}{3}$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $- 1$ by $3$ .

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

Step 2.5.2

Subtract $3$ from $2$ .

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

Step 2.6

Move the negative in front of the fraction.

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

Step 2.7

Simplify.

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Step 2.7.1

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

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

Step 2.7.2

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

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

Step 3

Find the second derivative of the function.

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Step 3.1

Since $\frac{2}{3}$ is constant with respect to $x$ , the derivative of $\frac{2}{3 x^{\frac{1}{3}}}$ with respect to $x$ is $\frac{2}{3} \frac{d}{d x} [\frac{1}{x^{\frac{1}{3}}}]$ .

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

Step 3.2

Apply basic rules of exponents.

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Step 3.2.1

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

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

Step 3.2.2

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

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Step 3.2.2.1

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

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

Step 3.2.2.2

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

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

Step 3.2.2.3

Move the negative in front of the fraction.

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Combine the numerators over the common denominator.

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

Step 3.7

Simplify the numerator.

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Step 3.7.1

Multiply $- 1$ by $3$ .

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

Step 3.7.2

Subtract $3$ from $- 1$ .

$f'' (x) = \frac{2}{3} \cdot (- \frac{1}{3} x^{\frac{- 4}{3}})$

Step 3.8

Move the negative in front of the fraction.

$f'' (x) = \frac{2}{3} \cdot (- \frac{1}{3} x^{- \frac{4}{3}})$

Step 3.9

Combine $x^{- \frac{4}{3}}$ and $\frac{1}{3}$ .

$f'' (x) = \frac{2}{3} \cdot (- \frac{x^{- \frac{4}{3}}}{3})$

Step 3.10

Multiply $\frac{2}{3}$ by $\frac{x^{- \frac{4}{3}}}{3}$ .

$f'' (x) = - \frac{2 x^{- \frac{4}{3}}}{3 \cdot 3}$

Step 3.11

Multiply.

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Step 3.11.1

Multiply $3$ by $3$ .

$f'' (x) = - \frac{2 x^{- \frac{4}{3}}}{9}$

Step 3.11.2

Move $x^{- \frac{4}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = - \frac{2}{9 x^{\frac{4}{3}}}$

Step 4

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

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

Step 5

Find the first derivative.

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Step 5.1

Find the first derivative.

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Step 5.1.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{2}{3}$ .

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

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.1.4

Combine the numerators over the common denominator.

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

Step 5.1.5

Simplify the numerator.

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Step 5.1.5.1

Multiply $- 1$ by $3$ .

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

Step 5.1.5.2

Subtract $3$ from $2$ .

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

Step 5.1.6

Move the negative in front of the fraction.

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

Step 5.1.7

Simplify.

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Step 5.1.7.1

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

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

Step 5.1.7.2

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

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

Step 5.2

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

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

Step 6

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

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Step 6.1

Set the first derivative equal to $0$ .

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

Step 6.2

Set the numerator equal to zero.

$2 = 0$

Step 6.3

Since $2 \neq 0$ , there are no solutions.

No solution

Step 7

Find the values where the derivative is undefined.

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Step 7.1

Convert expressions with fractional exponents to radicals.

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Step 7.1.1

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

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

Step 7.1.2

Anything raised to $1$ is the base itself.

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

Step 7.2

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 7.3

Solve for $x$ .

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Step 7.3.1

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 7.3.2

Simplify each side of the equation.

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Step 7.3.2.1

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 7.3.2.2

Simplify the left side.

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Step 7.3.2.2.1

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

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Step 7.3.2.2.1.1

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

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

Step 7.3.2.2.1.2

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

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

Step 7.3.2.2.1.3

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

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Step 7.3.2.2.1.3.1

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

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

Step 7.3.2.2.1.3.2

Cancel the common factor of $3$ .

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Step 7.3.2.2.1.3.2.1

Cancel the common factor.

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

Step 7.3.2.2.1.3.2.2

Rewrite the expression.

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

Step 7.3.2.2.1.4

Simplify.

$27 x = 0^{3}$

Step 7.3.2.3

Simplify the right side.

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Step 7.3.2.3.1

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

$27 x = 0$

Step 7.3.3

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

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Step 7.3.3.1

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

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

Step 7.3.3.2

Simplify the left side.

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Step 7.3.3.2.1

Cancel the common factor of $27$ .

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Step 7.3.3.2.1.1

Cancel the common factor.

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

Step 7.3.3.2.1.2

Divide $x$ by $1$ .

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

Step 7.3.3.3

Simplify the right side.

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Step 7.3.3.3.1

Divide $0$ by $27$ .

$x = 0$

Step 8

Critical points to evaluate.

$x = 0$

Step 9

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{2}{9 {(0)}^{\frac{4}{3}}}$

Step 10

Evaluate the second derivative.

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Step 10.1

Simplify the expression.

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Step 10.1.1

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

$- \frac{2}{9 {(0^{3})}^{\frac{4}{3}}}$

Step 10.1.2

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

$- \frac{2}{9 \cdot 0^{3 (\frac{4}{3})}}$

Step 10.2

Cancel the common factor of $3$ .

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Step 10.2.1

Cancel the common factor.

$- \frac{2}{9 \cdot 0^{3 (\frac{4}{3})}}$

Step 10.2.2

Rewrite the expression.

$- \frac{2}{9 \cdot 0^{4}}$

Step 10.3

Simplify the expression.

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Step 10.3.1

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

$- \frac{2}{9 \cdot 0}$

Step 10.3.2

Multiply $9$ by $0$ .

$- \frac{2}{0}$

Step 10.3.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$- \frac{2}{0}$

Step 10.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 11

Since there is at least one point with $0$ or undefined second derivative, apply the first derivative test.

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Step 11.1

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

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

Step 11.2

Substitute any number, such as $- 2$ , from the interval $(- \infty, 0)$ in the first derivative $\frac{2}{3 x^{\frac{1}{3}}}$ to check if the result is negative or positive.

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Step 11.2.1

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

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

Step 11.2.2

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

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

Step 11.3

Substitute any number, such as $2$ , from the interval $(0, \infty)$ in the first derivative $\frac{2}{3 x^{\frac{1}{3}}}$ to check if the result is negative or positive.

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Step 11.3.1

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

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

Step 11.3.2

Simplify the result.

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Step 11.3.2.1

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

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

Step 11.3.2.2

Multiply $2$ by $2^{- \frac{1}{3}}$ by adding the exponents.

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Step 11.3.2.2.1

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

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Step 11.3.2.2.1.1

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

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

Step 11.3.2.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 11.3.2.2.2

Write $1$ as a fraction with a common denominator.

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

Step 11.3.2.2.3

Combine the numerators over the common denominator.

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

Step 11.3.2.2.4

Subtract $1$ from $3$ .

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

Step 11.3.2.3

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

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

Step 11.4

Since the first derivative changed signs from negative to positive around $x = 0$ , then $x = 0$ is a local minimum.

$x = 0$ is a local minimum

Step 12