Calculus Examples

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

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

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Step 1.1.1.1

To apply the Chain Rule, set $u$ as $x^{2} - 1$ .

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

Step 1.1.1.2

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

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

Step 1.1.1.3

Replace all occurrences of $u$ with $x^{2} - 1$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

Combine the numerators over the common denominator.

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

Step 1.1.5

Simplify the numerator.

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Step 1.1.5.1

Multiply $- 1$ by $3$ .

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

Step 1.1.5.2

Subtract $3$ from $1$ .

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

Step 1.1.6

Combine fractions.

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Step 1.1.6.1

Move the negative in front of the fraction.

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

Step 1.1.6.2

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

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

Step 1.1.6.3

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

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

Step 1.1.7

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

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

Step 1.1.8

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

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

Step 1.1.9

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

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

Step 1.1.10

Combine fractions.

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Step 1.1.10.1

Add $2 x$ and $0$ .

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

Step 1.1.10.2

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

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

Step 1.1.10.3

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

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Set the numerator equal to zero.

$2 x = 0$

Step 2.3

Divide each term in $2 x = 0$ by $2$ and simplify.

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Step 2.3.1

Divide each term in $2 x = 0$ by $2$ .

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

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $2$ .

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Step 2.3.2.1.1

Cancel the common factor.

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

Step 2.3.2.1.2

Divide $x$ by $1$ .

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

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $0$ by $2$ .

$x = 0$

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.

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

Step 3.2

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

$3 \sqrt[3]{{(x^{2} - 1)}^{2}} = 0$

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.

${(3 \sqrt[3]{{(x^{2} - 1)}^{2}})}^{3} = 0^{3}$

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 $\sqrt[3]{{(x^{2} - 1)}^{2}}$ as ${(x^{2} - 1)}^{\frac{2}{3}}$ .

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

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

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

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Step 3.3.2.2.1.1

Apply the product rule to $3 {(x^{2} - 1)}^{\frac{2}{3}}$ .

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

Step 3.3.2.2.1.2

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

$27 {({(x^{2} - 1)}^{\frac{2}{3}})}^{3} = 0^{3}$

Step 3.3.2.2.1.3

Multiply the exponents in ${({(x^{2} - 1)}^{\frac{2}{3}})}^{3}$ .

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Step 3.3.2.2.1.3.1

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

$27 {(x^{2} - 1)}^{\frac{2}{3} \cdot 3} = 0^{3}$

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.

$27 {(x^{2} - 1)}^{\frac{2}{3} \cdot 3} = 0^{3}$

Step 3.3.2.2.1.3.2.2

Rewrite the expression.

$27 {(x^{2} - 1)}^{2} = 0^{3}$

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 $0$ .

$27 {(x^{2} - 1)}^{2} = 0$

Step 3.3.3

Solve for $x$ .

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Step 3.3.3.1

Factor the left side of the equation.

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Step 3.3.3.1.1

Rewrite $1$ as $1^{2}$ .

$27 {(x^{2} - 1^{2})}^{2} = 0$

Step 3.3.3.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$27 {((x + 1) (x - 1))}^{2} = 0$

Step 3.3.3.1.3

Factor.

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Step 3.3.3.1.3.1

Apply the product rule to $(x + 1) (x - 1)$ .

$27 ({(x + 1)}^{2} {(x - 1)}^{2}) = 0$

Step 3.3.3.1.3.2

Remove unnecessary parentheses.

$27 {(x + 1)}^{2} {(x - 1)}^{2} = 0$

Step 3.3.3.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

${(x + 1)}^{2} = 0$

${(x - 1)}^{2} = 0$

Step 3.3.3.3

Set ${(x + 1)}^{2}$ equal to $0$ and solve for $x$ .

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Step 3.3.3.3.1

Set ${(x + 1)}^{2}$ equal to $0$ .

${(x + 1)}^{2} = 0$

Step 3.3.3.3.2

Solve ${(x + 1)}^{2} = 0$ for $x$ .

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Step 3.3.3.3.2.1

Set the $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 3.3.3.3.2.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 3.3.3.4

Set ${(x - 1)}^{2}$ equal to $0$ and solve for $x$ .

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Step 3.3.3.4.1

Set ${(x - 1)}^{2}$ equal to $0$ .

${(x - 1)}^{2} = 0$

Step 3.3.3.4.2

Solve ${(x - 1)}^{2} = 0$ for $x$ .

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Step 3.3.3.4.2.1

Set the $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 3.3.3.4.2.2

Add $1$ to both sides of the equation.

$x = 1$

Step 3.3.3.5

The final solution is all the values that make $27 {(x + 1)}^{2} {(x - 1)}^{2} = 0$ true.

$x = - 1, 1$

Step 3.4

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = - 1, x = 1$

Step 4

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

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

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

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

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

Step 4.1.2.1.2

Subtract $1$ from $0$ .

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

Step 4.1.2.1.3

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

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

Step 4.1.2.1.4

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

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

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.

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

Step 4.1.2.2.2

Rewrite the expression.

${(- 1)}^{1}$

Step 4.1.2.3

Evaluate the exponent.

$- 1$

Step 4.2

Evaluate at $x = - 1$ .

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Step 4.2.1

Substitute $- 1$ for $x$ .

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

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify the expression.

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Step 4.2.2.1.1

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

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

Step 4.2.2.1.2

Subtract $1$ from $1$ .

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

Step 4.2.2.1.3

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

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

Step 4.2.2.1.4

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

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

Step 4.2.2.2

Cancel the common factor of $3$ .

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Step 4.2.2.2.1

Cancel the common factor.

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

Step 4.2.2.2.2

Rewrite the expression.

$0^{1}$

Step 4.2.2.3

Evaluate the exponent.

$0$

Step 4.3

Evaluate at $x = 1$ .

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Step 4.3.1

Substitute $1$ for $x$ .

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

Step 4.3.2

Simplify.

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Step 4.3.2.1

Simplify the expression.

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Step 4.3.2.1.1

One to any power is one.

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

Step 4.3.2.1.2

Subtract $1$ from $1$ .

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

Step 4.3.2.1.3

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

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

Step 4.3.2.1.4

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

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

Step 4.3.2.2

Cancel the common factor of $3$ .

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Step 4.3.2.2.1

Cancel the common factor.

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

Step 4.3.2.2.2

Rewrite the expression.

$0^{1}$

Step 4.3.2.3

Evaluate the exponent.

$0$

Step 4.4

List all of the points.

$(0, - 1), (- 1, 0), (1, 0)$

Step 5