Calculus Examples

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

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

Differentiate.

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Step 1.1.1.1

By the Sum Rule, the derivative of $1 - \frac{1}{x^{\frac{2}{3}}}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- \frac{1}{x^{\frac{2}{3}}}]$ .

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

Step 1.1.1.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- \frac{1}{x^{\frac{2}{3}}}]$

Step 1.1.2

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

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Step 1.1.2.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = - 1$ and $g (x) = \frac{1}{x^{\frac{2}{3}}}$ .

$0 - \frac{d}{d x} [\frac{1}{x^{\frac{2}{3}}}] + \frac{1}{x^{\frac{2}{3}}} \frac{d}{d x} [- 1]$

Step 1.1.2.2

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

$0 - \frac{d}{d x} [{(x^{\frac{2}{3}})}^{- 1}] + \frac{1}{x^{\frac{2}{3}}} \frac{d}{d x} [- 1]$

Step 1.1.2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{\frac{2}{3}}$ .

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Step 1.1.2.3.1

To apply the Chain Rule, set $u$ as $x^{\frac{2}{3}}$ .

$0 - (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{\frac{2}{3}}]) + \frac{1}{x^{\frac{2}{3}}} \frac{d}{d x} [- 1]$

Step 1.1.2.3.2

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

$0 - (- u^{- 2} \frac{d}{d x} [x^{\frac{2}{3}}]) + \frac{1}{x^{\frac{2}{3}}} \frac{d}{d x} [- 1]$

Step 1.1.2.3.3

Replace all occurrences of $u$ with $x^{\frac{2}{3}}$ .

$0 - (- {(x^{\frac{2}{3}})}^{- 2} \frac{d}{d x} [x^{\frac{2}{3}}]) + \frac{1}{x^{\frac{2}{3}}} \frac{d}{d x} [- 1]$

Step 1.1.2.4

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

$0 - (- {(x^{\frac{2}{3}})}^{- 2} (\frac{2}{3} x^{\frac{2}{3} - 1})) + \frac{1}{x^{\frac{2}{3}}} \frac{d}{d x} [- 1]$

Step 1.1.2.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$0 - (- {(x^{\frac{2}{3}})}^{- 2} (\frac{2}{3} x^{\frac{2}{3} - 1})) + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 1.1.2.6

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

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Step 1.1.2.6.1

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

$0 - (- x^{\frac{2}{3} \cdot - 2} (\frac{2}{3} x^{\frac{2}{3} - 1})) + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 1.1.2.6.2

Multiply $\frac{2}{3} \cdot - 2$ .

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Step 1.1.2.6.2.1

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

$0 - (- x^{\frac{2 \cdot - 2}{3}} (\frac{2}{3} x^{\frac{2}{3} - 1})) + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 1.1.2.6.2.2

Multiply $2$ by $- 2$ .

$0 - (- x^{\frac{- 4}{3}} (\frac{2}{3} x^{\frac{2}{3} - 1})) + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 1.1.2.6.3

Move the negative in front of the fraction.

$0 - (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{2}{3} - 1})) + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 1.1.2.7

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

$0 - (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{2}{3} - 1 \cdot \frac{3}{3}})) + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 1.1.2.8

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

$0 - (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}})) + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 1.1.2.9

Combine the numerators over the common denominator.

$0 - (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{2 - 1 \cdot 3}{3}})) + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 1.1.2.10

Simplify the numerator.

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Step 1.1.2.10.1

Multiply $- 1$ by $3$ .

$0 - (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{2 - 3}{3}})) + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 1.1.2.10.2

Subtract $3$ from $2$ .

$0 - (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{- 1}{3}})) + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 1.1.2.11

Move the negative in front of the fraction.

$0 - (- x^{- \frac{4}{3}} (\frac{2}{3} x^{- \frac{1}{3}})) + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 1.1.2.12

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

$0 - (- x^{- \frac{4}{3}} \frac{2 x^{- \frac{1}{3}}}{3}) + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 1.1.2.13

Combine $\frac{2 x^{- \frac{1}{3}}}{3}$ and $x^{- \frac{4}{3}}$ .

$0 - - \frac{2 x^{- \frac{1}{3}} x^{- \frac{4}{3}}}{3} + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 1.1.2.14

Multiply $x^{- \frac{1}{3}}$ by $x^{- \frac{4}{3}}$ by adding the exponents.

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Step 1.1.2.14.1

Move $x^{- \frac{4}{3}}$ .

$0 - - \frac{2 (x^{- \frac{4}{3}} x^{- \frac{1}{3}})}{3} + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 1.1.2.14.2

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

$0 - - \frac{2 x^{- \frac{4}{3} - \frac{1}{3}}}{3} + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 1.1.2.14.3

Combine the numerators over the common denominator.

$0 - - \frac{2 x^{\frac{- 4 - 1}{3}}}{3} + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 1.1.2.14.4

Subtract $1$ from $- 4$ .

$0 - - \frac{2 x^{\frac{- 5}{3}}}{3} + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 1.1.2.14.5

Move the negative in front of the fraction.

$0 - - \frac{2 x^{- \frac{5}{3}}}{3} + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 1.1.2.15

Move $x^{- \frac{5}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$0 - - \frac{2}{3 x^{\frac{5}{3}}} + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 1.1.2.16

Multiply $- 1$ by $- 1$ .

$0 + 1 \frac{2}{3 x^{\frac{5}{3}}} + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 1.1.2.17

Multiply $\frac{2}{3 x^{\frac{5}{3}}}$ by $1$ .

$0 + \frac{2}{3 x^{\frac{5}{3}}} + \frac{1}{x^{\frac{2}{3}}} \cdot 0$

Step 1.1.2.18

Multiply $\frac{1}{x^{\frac{2}{3}}}$ by $0$ .

$0 + \frac{2}{3 x^{\frac{5}{3}}} + 0$

Step 1.1.2.19

Add $\frac{2}{3 x^{\frac{5}{3}}}$ and $0$ .

$0 + \frac{2}{3 x^{\frac{5}{3}}}$

Step 1.1.3

Add $0$ and $\frac{2}{3 x^{\frac{5}{3}}}$ .

$f' (x) = \frac{2}{3 x^{\frac{5}{3}}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{2}{3 x^{\frac{5}{3}}}$ .

$\frac{2}{3 x^{\frac{5}{3}}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{2}{3 x^{\frac{5}{3}}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{2}{3 x^{\frac{5}{3}}} = 0$

Step 2.2

Set the numerator equal to zero.

$2 = 0$

Step 2.3

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

No solution

Step 3

Find the values where the derivative is undefined.

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Step 3.1

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

$\frac{2}{3 \sqrt[3]{x^{5}}}$

Step 3.2

Set the denominator in $\frac{2}{3 \sqrt[3]{x^{5}}}$ equal to $0$ to find where the expression is undefined.

$3 \sqrt[3]{x^{5}} = 0$

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.

${(3 \sqrt[3]{x^{5}})}^{3} = 0^{3}$

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 $\sqrt[3]{x^{5}}$ as $x^{\frac{5}{3}}$ .

${(3 x^{\frac{5}{3}})}^{3} = 0^{3}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Simplify ${(3 x^{\frac{5}{3}})}^{3}$ .

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Step 3.3.2.2.1.1

Apply the product rule to $3 x^{\frac{5}{3}}$ .

$3^{3} {(x^{\frac{5}{3}})}^{3} = 0^{3}$

Step 3.3.2.2.1.2

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

$27 {(x^{\frac{5}{3}})}^{3} = 0^{3}$

Step 3.3.2.2.1.3

Multiply the exponents in ${(x^{\frac{5}{3}})}^{3}$ .

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Step 3.3.2.2.1.3.1

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

$27 x^{\frac{5}{3} \cdot 3} = 0^{3}$

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.

$27 x^{\frac{5}{3} \cdot 3} = 0^{3}$

Step 3.3.2.2.1.3.2.2

Rewrite the expression.

$27 x^{5} = 0^{3}$

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 $0$ .

$27 x^{5} = 0$

Step 3.3.3

Solve for $x$ .

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Step 3.3.3.1

Divide each term in $27 x^{5} = 0$ by $27$ and simplify.

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Step 3.3.3.1.1

Divide each term in $27 x^{5} = 0$ by $27$ .

$\frac{27 x^{5}}{27} = \frac{0}{27}$

Step 3.3.3.1.2

Simplify the left side.

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Step 3.3.3.1.2.1

Cancel the common factor of $27$ .

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Step 3.3.3.1.2.1.1

Cancel the common factor.

$\frac{27 x^{5}}{27} = \frac{0}{27}$

Step 3.3.3.1.2.1.2

Divide $x^{5}$ by $1$ .

$x^{5} = \frac{0}{27}$

Step 3.3.3.1.3

Simplify the right side.

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Step 3.3.3.1.3.1

Divide $0$ by $27$ .

$x^{5} = 0$

Step 3.3.3.2

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

$x = \sqrt[5]{0}$

Step 3.3.3.3

Simplify $\sqrt[5]{0}$ .

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Step 3.3.3.3.1

Rewrite $0$ as $0^{5}$ .

$x = \sqrt[5]{0^{5}}$

Step 3.3.3.3.2

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

$x = 0$

Step 4

Evaluate $1 - \frac{1}{x^{\frac{2}{3}}}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 0$ .

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Step 4.1.1

Substitute $0$ for $x$ .

$1 - \frac{1}{{(0)}^{\frac{2}{3}}}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify the expression.

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Step 4.1.2.1.1

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

$1 - \frac{1}{{(0^{3})}^{\frac{2}{3}}}$

Step 4.1.2.1.2

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

$1 - \frac{1}{0^{3 (\frac{2}{3})}}$

Step 4.1.2.2

Cancel the common factor of $3$ .

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Step 4.1.2.2.1

Cancel the common factor.

$1 - \frac{1}{0^{3 (\frac{2}{3})}}$

Step 4.1.2.2.2

Rewrite the expression.

$1 - \frac{1}{0^{2}}$

Step 4.1.2.3

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

$1 - \frac{1}{0}$

Step 4.1.2.4

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

Undefined

Step 5

There are no values of $x$ in the domain of the original problem where the derivative is $0$ or undefined.

No critical points found