Calculus Examples

$y = x^{3} - 4 x$

Step 1

Set $y$ as a function of $x$ .

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

Step 2

Find the derivative.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $x^{3} - 4 x$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 4 x]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 4 x]$

Step 2.1.2

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

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

Step 2.2

Evaluate $\frac{d}{d x} [- 4 x]$ .

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Step 2.2.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x$ with respect to $x$ is $- 4 \frac{d}{d x} [x]$ .

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $- 4$ by $1$ .

$3 x^{2} - 4$

Step 3

Set the derivative equal to $0$ then solve the equation $3 x^{2} - 4 = 0$ .

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Step 3.1

Add $4$ to both sides of the equation.

$3 x^{2} = 4$

Step 3.2

Divide each term in $3 x^{2} = 4$ by $3$ and simplify.

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Step 3.2.1

Divide each term in $3 x^{2} = 4$ by $3$ .

$\frac{3 x^{2}}{3} = \frac{4}{3}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $3$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{4}{3}$

Step 3.2.2.1.2

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

$x^{2} = \frac{4}{3}$

Step 3.3

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

$x = \pm \sqrt{\frac{4}{3}}$

Step 3.4

Simplify $\pm \sqrt{\frac{4}{3}}$ .

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Step 3.4.1

Rewrite $\sqrt{\frac{4}{3}}$ as $\frac{\sqrt{4}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{4}}{\sqrt{3}}$

Step 3.4.2

Simplify the numerator.

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Step 3.4.2.1

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

$x = \pm \frac{\sqrt{2^{2}}}{\sqrt{3}}$

Step 3.4.2.2

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

$x = \pm \frac{2}{\sqrt{3}}$

Step 3.4.3

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 3.4.4

Combine and simplify the denominator.

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Step 3.4.4.1

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 3.4.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 3.4.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 3.4.4.4

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

$x = \pm \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 3.4.4.5

Add $1$ and $1$ .

$x = \pm \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 3.4.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.4.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = \pm \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 3.4.4.6.2

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

$x = \pm \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 3.4.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 3.4.4.6.4

Cancel the common factor of $2$ .

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Step 3.4.4.6.4.1

Cancel the common factor.

$x = \pm \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 3.4.4.6.4.2

Rewrite the expression.

$x = \pm \frac{2 \sqrt{3}}{3^{1}}$

Step 3.4.4.6.5

Evaluate the exponent.

$x = \pm \frac{2 \sqrt{3}}{3}$

Step 3.5

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

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Step 3.5.1

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

$x = \frac{2 \sqrt{3}}{3}$

Step 3.5.2

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

$x = - \frac{2 \sqrt{3}}{3}$

Step 3.5.3

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

$x = \frac{2 \sqrt{3}}{3}, - \frac{2 \sqrt{3}}{3}$

Step 4

Solve the original function $f (x) = x^{3} - 4 x$ at $x = \frac{2 \sqrt{3}}{3}$ .

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Step 4.1

Replace the variable $x$ with $\frac{2 \sqrt{3}}{3}$ in the expression.

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

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.2.1.1.1

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

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

Step 4.2.1.1.2

Apply the product rule to $2 \sqrt{3}$ .

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

Step 4.2.1.2

Simplify the numerator.

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Step 4.2.1.2.1

Raise $2$ to the power of $3$ .

$f (\frac{2 \sqrt{3}}{3}) = \frac{8 {\sqrt{3}}^{3}}{3^{3}} - 4 \frac{2 \sqrt{3}}{3}$

Step 4.2.1.2.2

Rewrite ${\sqrt{3}}^{3}$ as $\sqrt{3^{3}}$ .

$f (\frac{2 \sqrt{3}}{3}) = \frac{8 \sqrt{3^{3}}}{3^{3}} - 4 \frac{2 \sqrt{3}}{3}$

Step 4.2.1.2.3

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

$f (\frac{2 \sqrt{3}}{3}) = \frac{8 \sqrt{27}}{3^{3}} - 4 \frac{2 \sqrt{3}}{3}$

Step 4.2.1.2.4

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 4.2.1.2.4.1

Factor $9$ out of $27$ .

$f (\frac{2 \sqrt{3}}{3}) = \frac{8 \sqrt{9 (3)}}{3^{3}} - 4 \frac{2 \sqrt{3}}{3}$

Step 4.2.1.2.4.2

Rewrite $9$ as $3^{2}$ .

$f (\frac{2 \sqrt{3}}{3}) = \frac{8 \sqrt{3^{2} \cdot 3}}{3^{3}} - 4 \frac{2 \sqrt{3}}{3}$

Step 4.2.1.2.5

Pull terms out from under the radical.

$f (\frac{2 \sqrt{3}}{3}) = \frac{8 \cdot (3 \sqrt{3})}{3^{3}} - 4 \frac{2 \sqrt{3}}{3}$

Step 4.2.1.2.6

Multiply $8$ by $3$ .

$f (\frac{2 \sqrt{3}}{3}) = \frac{24 \sqrt{3}}{3^{3}} - 4 \frac{2 \sqrt{3}}{3}$

Step 4.2.1.3

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

$f (\frac{2 \sqrt{3}}{3}) = \frac{24 \sqrt{3}}{27} - 4 \frac{2 \sqrt{3}}{3}$

Step 4.2.1.4

Cancel the common factor of $24$ and $27$ .

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Step 4.2.1.4.1

Factor $3$ out of $24 \sqrt{3}$ .

$f (\frac{2 \sqrt{3}}{3}) = \frac{3 (8 \sqrt{3})}{27} - 4 \frac{2 \sqrt{3}}{3}$

Step 4.2.1.4.2

Cancel the common factors.

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Step 4.2.1.4.2.1

Factor $3$ out of $27$ .

$f (\frac{2 \sqrt{3}}{3}) = \frac{3 (8 \sqrt{3})}{3 (9)} - 4 \frac{2 \sqrt{3}}{3}$

Step 4.2.1.4.2.2

Cancel the common factor.

$f (\frac{2 \sqrt{3}}{3}) = \frac{3 (8 \sqrt{3})}{3 \cdot 9} - 4 \frac{2 \sqrt{3}}{3}$

Step 4.2.1.4.2.3

Rewrite the expression.

$f (\frac{2 \sqrt{3}}{3}) = \frac{8 \sqrt{3}}{9} - 4 \frac{2 \sqrt{3}}{3}$

Step 4.2.1.5

Multiply $- 4 \frac{2 \sqrt{3}}{3}$ .

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Step 4.2.1.5.1

Combine $- 4$ and $\frac{2 \sqrt{3}}{3}$ .

$f (\frac{2 \sqrt{3}}{3}) = \frac{8 \sqrt{3}}{9} + \frac{- 4 (2 \sqrt{3})}{3}$

Step 4.2.1.5.2

Multiply $2$ by $- 4$ .

$f (\frac{2 \sqrt{3}}{3}) = \frac{8 \sqrt{3}}{9} + \frac{- 8 \sqrt{3}}{3}$

Step 4.2.1.6

Move the negative in front of the fraction.

$f (\frac{2 \sqrt{3}}{3}) = \frac{8 \sqrt{3}}{9} - \frac{8 \sqrt{3}}{3}$

Step 4.2.2

To write $- \frac{8 \sqrt{3}}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (\frac{2 \sqrt{3}}{3}) = \frac{8 \sqrt{3}}{9} - \frac{8 \sqrt{3}}{3} \cdot \frac{3}{3}$

Step 4.2.3

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.3.1

Multiply $\frac{8 \sqrt{3}}{3}$ by $\frac{3}{3}$ .

$f (\frac{2 \sqrt{3}}{3}) = \frac{8 \sqrt{3}}{9} - \frac{8 \sqrt{3} \cdot 3}{3 \cdot 3}$

Step 4.2.3.2

Multiply $3$ by $3$ .

$f (\frac{2 \sqrt{3}}{3}) = \frac{8 \sqrt{3}}{9} - \frac{8 \sqrt{3} \cdot 3}{9}$

Step 4.2.4

Combine the numerators over the common denominator.

$f (\frac{2 \sqrt{3}}{3}) = \frac{8 \sqrt{3} - 8 \sqrt{3} \cdot 3}{9}$

Step 4.2.5

Simplify the numerator.

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Step 4.2.5.1

Multiply $3$ by $- 8$ .

$f (\frac{2 \sqrt{3}}{3}) = \frac{8 \sqrt{3} - 24 \sqrt{3}}{9}$

Step 4.2.5.2

Subtract $24 \sqrt{3}$ from $8 \sqrt{3}$ .

$f (\frac{2 \sqrt{3}}{3}) = \frac{- 16 \sqrt{3}}{9}$

Step 4.2.6

Move the negative in front of the fraction.

$f (\frac{2 \sqrt{3}}{3}) = - \frac{16 \sqrt{3}}{9}$

Step 4.2.7

The final answer is $- \frac{16 \sqrt{3}}{9}$ .

$- \frac{16 \sqrt{3}}{9}$

Step 5

Solve the original function $f (x) = x^{3} - 4 x$ at $x = - \frac{2 \sqrt{3}}{3}$ .

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Step 5.1

Replace the variable $x$ with $- \frac{2 \sqrt{3}}{3}$ in the expression.

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

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.2.1.1.1

Apply the product rule to $- \frac{2 \sqrt{3}}{3}$ .

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

Step 5.2.1.1.2

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

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

Step 5.2.1.1.3

Apply the product rule to $2 \sqrt{3}$ .

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

Step 5.2.1.2

Raise $- 1$ to the power of $3$ .

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

Step 5.2.1.3

Simplify the numerator.

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Step 5.2.1.3.1

Raise $2$ to the power of $3$ .

$f (- \frac{2 \sqrt{3}}{3}) = - \frac{8 {\sqrt{3}}^{3}}{3^{3}} - 4 (- \frac{2 \sqrt{3}}{3})$

Step 5.2.1.3.2

Rewrite ${\sqrt{3}}^{3}$ as $\sqrt{3^{3}}$ .

$f (- \frac{2 \sqrt{3}}{3}) = - \frac{8 \sqrt{3^{3}}}{3^{3}} - 4 (- \frac{2 \sqrt{3}}{3})$

Step 5.2.1.3.3

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

$f (- \frac{2 \sqrt{3}}{3}) = - \frac{8 \sqrt{27}}{3^{3}} - 4 (- \frac{2 \sqrt{3}}{3})$

Step 5.2.1.3.4

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 5.2.1.3.4.1

Factor $9$ out of $27$ .

$f (- \frac{2 \sqrt{3}}{3}) = - \frac{8 \sqrt{9 (3)}}{3^{3}} - 4 (- \frac{2 \sqrt{3}}{3})$

Step 5.2.1.3.4.2

Rewrite $9$ as $3^{2}$ .

$f (- \frac{2 \sqrt{3}}{3}) = - \frac{8 \sqrt{3^{2} \cdot 3}}{3^{3}} - 4 (- \frac{2 \sqrt{3}}{3})$

Step 5.2.1.3.5

Pull terms out from under the radical.

$f (- \frac{2 \sqrt{3}}{3}) = - \frac{8 \cdot (3 \sqrt{3})}{3^{3}} - 4 (- \frac{2 \sqrt{3}}{3})$

Step 5.2.1.3.6

Multiply $8$ by $3$ .

$f (- \frac{2 \sqrt{3}}{3}) = - \frac{24 \sqrt{3}}{3^{3}} - 4 (- \frac{2 \sqrt{3}}{3})$

Step 5.2.1.4

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

$f (- \frac{2 \sqrt{3}}{3}) = - \frac{24 \sqrt{3}}{27} - 4 (- \frac{2 \sqrt{3}}{3})$

Step 5.2.1.5

Cancel the common factor of $24$ and $27$ .

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Step 5.2.1.5.1

Factor $3$ out of $24 \sqrt{3}$ .

$f (- \frac{2 \sqrt{3}}{3}) = - \frac{3 (8 \sqrt{3})}{27} - 4 (- \frac{2 \sqrt{3}}{3})$

Step 5.2.1.5.2

Cancel the common factors.

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Step 5.2.1.5.2.1

Factor $3$ out of $27$ .

$f (- \frac{2 \sqrt{3}}{3}) = - \frac{3 (8 \sqrt{3})}{3 (9)} - 4 (- \frac{2 \sqrt{3}}{3})$

Step 5.2.1.5.2.2

Cancel the common factor.

$f (- \frac{2 \sqrt{3}}{3}) = - \frac{3 (8 \sqrt{3})}{3 \cdot 9} - 4 (- \frac{2 \sqrt{3}}{3})$

Step 5.2.1.5.2.3

Rewrite the expression.

$f (- \frac{2 \sqrt{3}}{3}) = - \frac{8 \sqrt{3}}{9} - 4 (- \frac{2 \sqrt{3}}{3})$

Step 5.2.1.6

Multiply $- 4 (- \frac{2 \sqrt{3}}{3})$ .

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Step 5.2.1.6.1

Multiply $- 1$ by $- 4$ .

$f (- \frac{2 \sqrt{3}}{3}) = - \frac{8 \sqrt{3}}{9} + 4 (\frac{2 \sqrt{3}}{3})$

Step 5.2.1.6.2

Combine $4$ and $\frac{2 \sqrt{3}}{3}$ .

$f (- \frac{2 \sqrt{3}}{3}) = - \frac{8 \sqrt{3}}{9} + \frac{4 (2 \sqrt{3})}{3}$

Step 5.2.1.6.3

Multiply $2$ by $4$ .

$f (- \frac{2 \sqrt{3}}{3}) = - \frac{8 \sqrt{3}}{9} + \frac{8 \sqrt{3}}{3}$

Step 5.2.2

To write $\frac{8 \sqrt{3}}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (- \frac{2 \sqrt{3}}{3}) = - \frac{8 \sqrt{3}}{9} + \frac{8 \sqrt{3}}{3} \cdot \frac{3}{3}$

Step 5.2.3

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.2.3.1

Multiply $\frac{8 \sqrt{3}}{3}$ by $\frac{3}{3}$ .

$f (- \frac{2 \sqrt{3}}{3}) = - \frac{8 \sqrt{3}}{9} + \frac{8 \sqrt{3} \cdot 3}{3 \cdot 3}$

Step 5.2.3.2

Multiply $3$ by $3$ .

$f (- \frac{2 \sqrt{3}}{3}) = - \frac{8 \sqrt{3}}{9} + \frac{8 \sqrt{3} \cdot 3}{9}$

Step 5.2.4

Combine the numerators over the common denominator.

$f (- \frac{2 \sqrt{3}}{3}) = \frac{- 8 \sqrt{3} + 8 \sqrt{3} \cdot 3}{9}$

Step 5.2.5

Simplify the numerator.

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Step 5.2.5.1

Multiply $3$ by $8$ .

$f (- \frac{2 \sqrt{3}}{3}) = \frac{- 8 \sqrt{3} + 24 \sqrt{3}}{9}$

Step 5.2.5.2

Add $- 8 \sqrt{3}$ and $24 \sqrt{3}$ .

$f (- \frac{2 \sqrt{3}}{3}) = \frac{16 \sqrt{3}}{9}$

Step 5.2.6

The final answer is $\frac{16 \sqrt{3}}{9}$ .

$\frac{16 \sqrt{3}}{9}$

Step 6

The horizontal tangent lines on function $f (x) = x^{3} - 4 x$ are $y = - \frac{16 \sqrt{3}}{9}, y = \frac{16 \sqrt{3}}{9}$ .

$y = - \frac{16 \sqrt{3}}{9}, y = \frac{16 \sqrt{3}}{9}$

Step 7