Calculus Examples

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

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 using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{2}{3}}$ and $g (x) = x^{2} - 4$ .

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Step 1.1.1.1

To apply the Chain Rule, set $u$ as $x^{2} - 4$ .

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

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

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

Step 1.1.1.3

Replace all occurrences of $u$ with $x^{2} - 4$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

Combine the numerators over the common denominator.

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

Step 1.1.5

Simplify the numerator.

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Step 1.1.5.1

Multiply $- 1$ by $3$ .

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

Step 1.1.5.2

Subtract $3$ from $2$ .

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

Step 1.1.6

Combine fractions.

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Step 1.1.6.1

Move the negative in front of the fraction.

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

Step 1.1.6.2

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

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

Step 1.1.6.3

Move ${(x^{2} - 4)}^{- \frac{1}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.1.7

By the Sum Rule, the derivative of $x^{2} - 4$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4]$ .

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

Step 1.1.8

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

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

Step 1.1.9

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

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

Step 1.1.10

Combine fractions.

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Step 1.1.10.1

Add $2 x$ and $0$ .

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

Step 1.1.10.2

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

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

Step 1.1.10.3

Multiply $2$ by $2$ .

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

Step 1.1.10.4

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

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

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{4 x}{3 {(x^{2} - 4)}^{\frac{1}{3}}}$ .

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

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{4 x}{3 {(x^{2} - 4)}^{\frac{1}{3}}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Set the numerator equal to zero.

$4 x = 0$

Step 2.3

Divide each term in $4 x = 0$ by $4$ and simplify.

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Step 2.3.1

Divide each term in $4 x = 0$ by $4$ .

$\frac{4 x}{4} = \frac{0}{4}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $4$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{0}{4}$

Step 2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{4}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $0$ by $4$ .

$x = 0$

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.

$\frac{4 x}{3 \sqrt[3]{{(x^{2} - 4)}^{1}}}$

Step 3.1.2

Anything raised to $1$ is the base itself.

$\frac{4 x}{3 \sqrt[3]{x^{2} - 4}}$

Step 3.2

Set the denominator in $\frac{4 x}{3 \sqrt[3]{x^{2} - 4}}$ equal to $0$ to find where the expression is undefined.

$3 \sqrt[3]{x^{2} - 4} = 0$

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.

${(3 \sqrt[3]{x^{2} - 4})}^{3} = 0^{3}$

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 $\sqrt[3]{x^{2} - 4}$ as ${(x^{2} - 4)}^{\frac{1}{3}}$ .

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

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Simplify ${(3 {(x^{2} - 4)}^{\frac{1}{3}})}^{3}$ .

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Step 3.3.2.2.1.1

Apply the product rule to $3 {(x^{2} - 4)}^{\frac{1}{3}}$ .

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

Step 3.3.2.2.1.2

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

$27 {({(x^{2} - 4)}^{\frac{1}{3}})}^{3} = 0^{3}$

Step 3.3.2.2.1.3

Multiply the exponents in ${({(x^{2} - 4)}^{\frac{1}{3}})}^{3}$ .

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Step 3.3.2.2.1.3.1

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

$27 {(x^{2} - 4)}^{\frac{1}{3} \cdot 3} = 0^{3}$

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.

$27 {(x^{2} - 4)}^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 3.3.2.2.1.3.2.2

Rewrite the expression.

$27 {(x^{2} - 4)}^{1} = 0^{3}$

Step 3.3.2.2.1.4

Simplify.

$27 (x^{2} - 4) = 0^{3}$

Step 3.3.2.2.1.5

Apply the distributive property.

$27 x^{2} + 27 \cdot - 4 = 0^{3}$

Step 3.3.2.2.1.6

Multiply $27$ by $- 4$ .

$27 x^{2} - 108 = 0^{3}$

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 $0$ .

$27 x^{2} - 108 = 0$

Step 3.3.3

Solve for $x$ .

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Step 3.3.3.1

Add $108$ to both sides of the equation.

$27 x^{2} = 108$

Step 3.3.3.2

Divide each term in $27 x^{2} = 108$ by $27$ and simplify.

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Step 3.3.3.2.1

Divide each term in $27 x^{2} = 108$ by $27$ .

$\frac{27 x^{2}}{27} = \frac{108}{27}$

Step 3.3.3.2.2

Simplify the left side.

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Step 3.3.3.2.2.1

Cancel the common factor of $27$ .

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Step 3.3.3.2.2.1.1

Cancel the common factor.

$\frac{27 x^{2}}{27} = \frac{108}{27}$

Step 3.3.3.2.2.1.2

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

$x^{2} = \frac{108}{27}$

Step 3.3.3.2.3

Simplify the right side.

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Step 3.3.3.2.3.1

Divide $108$ by $27$ .

$x^{2} = 4$

Step 3.3.3.3

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

$x = \pm \sqrt{4}$

Step 3.3.3.4

Simplify $\pm \sqrt{4}$ .

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Step 3.3.3.4.1

Rewrite $4$ as $2^{2}$ .

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

Step 3.3.3.4.2

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

$x = \pm 2$

Step 3.3.3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.3.3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 2$

Step 3.3.3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 2$

Step 3.3.3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2, - 2$

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 $0$ .

$x = - 2, x = 2$

Step 4

Evaluate ${(x^{2} - 4)}^{\frac{2}{3}}$ 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 $x$ .

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

Step 4.1.2

Simplify.

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Step 4.1.2.1

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

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

Step 4.1.2.2

Subtract $4$ from $0$ .

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

Step 4.2

Evaluate at $x = - 2$ .

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Step 4.2.1

Substitute $- 2$ for $x$ .

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

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

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

Step 4.2.2.1.2

Subtract $4$ from $4$ .

$0^{\frac{2}{3}}$

Step 4.2.2.1.3

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

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

Step 4.2.2.1.4

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

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

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.

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

Step 4.2.2.2.2

Rewrite the expression.

$0^{2}$

Step 4.2.2.3

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

$0$

Step 4.3

Evaluate at $x = 2$ .

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Step 4.3.1

Substitute $2$ for $x$ .

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

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

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

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

Step 4.3.2.1.2

Subtract $4$ from $4$ .

$0^{\frac{2}{3}}$

Step 4.3.2.1.3

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

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

Step 4.3.2.1.4

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

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

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.

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

Step 4.3.2.2.2

Rewrite the expression.

$0^{2}$

Step 4.3.2.3

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

$0$

Step 4.4

List all of the points.

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

Step 5