Calculus Examples

$4 x^{3} - 16 x$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1

By the Sum Rule, the derivative of $4 x^{3} - 16 x$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 16 x]$ .

$\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 16 x]$

Step 1.1.2

Evaluate $\frac{d}{d x} [4 x^{3}]$ .

Tap for more steps...

Step 1.1.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3}$ with respect to $x$ is $4 \frac{d}{d x} [x^{3}]$ .

$4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 16 x]$

Step 1.1.2.2

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

$4 (3 x^{2}) + \frac{d}{d x} [- 16 x]$

Step 1.1.2.3

Multiply $3$ by $4$ .

$12 x^{2} + \frac{d}{d x} [- 16 x]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 16 x]$ .

Tap for more steps...

Step 1.1.3.1

Since $- 16$ is constant with respect to $x$ , the derivative of $- 16 x$ with respect to $x$ is $- 16 \frac{d}{d x} [x]$ .

$12 x^{2} - 16 \frac{d}{d x} [x]$

Step 1.1.3.2

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

$12 x^{2} - 16 \cdot 1$

Step 1.1.3.3

Multiply $- 16$ by $1$ .

$f' (x) = 12 x^{2} - 16$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $12 x^{2} - 16$ .

$12 x^{2} - 16$

Step 2

Set the first derivative equal to $0$ then solve the equation $12 x^{2} - 16 = 0$ .

Tap for more steps...

Step 2.1

Set the first derivative equal to $0$ .

$12 x^{2} - 16 = 0$

Step 2.2

Add $16$ to both sides of the equation.

$12 x^{2} = 16$

Step 2.3

Divide each term in $12 x^{2} = 16$ by $12$ and simplify.

Tap for more steps...

Step 2.3.1

Divide each term in $12 x^{2} = 16$ by $12$ .

$\frac{12 x^{2}}{12} = \frac{16}{12}$

Step 2.3.2

Simplify the left side.

Tap for more steps...

Step 2.3.2.1

Cancel the common factor of $12$ .

Tap for more steps...

Step 2.3.2.1.1

Cancel the common factor.

$\frac{12 x^{2}}{12} = \frac{16}{12}$

Step 2.3.2.1.2

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

$x^{2} = \frac{16}{12}$

Step 2.3.3

Simplify the right side.

Tap for more steps...

Step 2.3.3.1

Cancel the common factor of $16$ and $12$ .

Tap for more steps...

Step 2.3.3.1.1

Factor $4$ out of $16$ .

$x^{2} = \frac{4 (4)}{12}$

Step 2.3.3.1.2

Cancel the common factors.

Tap for more steps...

Step 2.3.3.1.2.1

Factor $4$ out of $12$ .

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

Step 2.3.3.1.2.2

Cancel the common factor.

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

Step 2.3.3.1.2.3

Rewrite the expression.

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

Step 2.4

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 2.5

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

Tap for more steps...

Step 2.5.1

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

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

Step 2.5.2

Simplify the numerator.

Tap for more steps...

Step 2.5.2.1

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

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

Step 2.5.2.2

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

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

Step 2.5.3

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

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

Step 2.5.4

Combine and simplify the denominator.

Tap for more steps...

Step 2.5.4.1

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

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

Step 2.5.4.2

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

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

Step 2.5.4.3

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

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

Step 2.5.4.4

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 2.5.4.5

Add $1$ and $1$ .

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

Step 2.5.4.6

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

Tap for more steps...

Step 2.5.4.6.1

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 2.5.4.6.2

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 2.5.4.6.3

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

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

Step 2.5.4.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.5.4.6.4.1

Cancel the common factor.

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

Step 2.5.4.6.4.2

Rewrite the expression.

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

Step 2.5.4.6.5

Evaluate the exponent.

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

Step 2.6

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

Tap for more steps...

Step 2.6.1

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

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

Step 2.6.2

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

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

Step 2.6.3

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 3

Find the values where the derivative is undefined.

Tap for more steps...

Step 3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

Evaluate $4 x^{3} - 16 x$ at each $x$ value where the derivative is $0$ or undefined.

Tap for more steps...

Step 4.1

Evaluate at $x = \frac{2 \sqrt{3}}{3}$ .

Tap for more steps...

Step 4.1.1

Substitute $\frac{2 \sqrt{3}}{3}$ for $x$ .

$4 {(\frac{2 \sqrt{3}}{3})}^{3} - 16 \frac{2 \sqrt{3}}{3}$

Step 4.1.2

Simplify.

Tap for more steps...

Step 4.1.2.1

Simplify each term.

Tap for more steps...

Step 4.1.2.1.1

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

Tap for more steps...

Step 4.1.2.1.1.1

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

$4 \frac{{(2 \sqrt{3})}^{3}}{3^{3}} - 16 \frac{2 \sqrt{3}}{3}$

Step 4.1.2.1.1.2

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

$4 \frac{2^{3} {\sqrt{3}}^{3}}{3^{3}} - 16 \frac{2 \sqrt{3}}{3}$

Step 4.1.2.1.2

Simplify the numerator.

Tap for more steps...

Step 4.1.2.1.2.1

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

$4 \frac{8 {\sqrt{3}}^{3}}{3^{3}} - 16 \frac{2 \sqrt{3}}{3}$

Step 4.1.2.1.2.2

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

$4 \frac{8 \sqrt{3^{3}}}{3^{3}} - 16 \frac{2 \sqrt{3}}{3}$

Step 4.1.2.1.2.3

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

$4 \frac{8 \sqrt{27}}{3^{3}} - 16 \frac{2 \sqrt{3}}{3}$

Step 4.1.2.1.2.4

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

Tap for more steps...

Step 4.1.2.1.2.4.1

Factor $9$ out of $27$ .

$4 \frac{8 \sqrt{9 (3)}}{3^{3}} - 16 \frac{2 \sqrt{3}}{3}$

Step 4.1.2.1.2.4.2

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

$4 \frac{8 \sqrt{3^{2} \cdot 3}}{3^{3}} - 16 \frac{2 \sqrt{3}}{3}$

Step 4.1.2.1.2.5

Pull terms out from under the radical.

$4 \frac{8 \cdot 3 \sqrt{3}}{3^{3}} - 16 \frac{2 \sqrt{3}}{3}$

Step 4.1.2.1.2.6

Multiply $8$ by $3$ .

$4 \frac{24 \sqrt{3}}{3^{3}} - 16 \frac{2 \sqrt{3}}{3}$

Step 4.1.2.1.3

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

$4 \frac{24 \sqrt{3}}{27} - 16 \frac{2 \sqrt{3}}{3}$

Step 4.1.2.1.4

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

Tap for more steps...

Step 4.1.2.1.4.1

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

$4 \frac{3 (8 \sqrt{3})}{27} - 16 \frac{2 \sqrt{3}}{3}$

Step 4.1.2.1.4.2

Cancel the common factors.

Tap for more steps...

Step 4.1.2.1.4.2.1

Factor $3$ out of $27$ .

$4 \frac{3 (8 \sqrt{3})}{3 (9)} - 16 \frac{2 \sqrt{3}}{3}$

Step 4.1.2.1.4.2.2

Cancel the common factor.

$4 \frac{3 (8 \sqrt{3})}{3 \cdot 9} - 16 \frac{2 \sqrt{3}}{3}$

Step 4.1.2.1.4.2.3

Rewrite the expression.

$4 \frac{8 \sqrt{3}}{9} - 16 \frac{2 \sqrt{3}}{3}$

Step 4.1.2.1.5

Multiply $4 \frac{8 \sqrt{3}}{9}$ .

Tap for more steps...

Step 4.1.2.1.5.1

Combine $4$ and $\frac{8 \sqrt{3}}{9}$ .

$\frac{4 (8 \sqrt{3})}{9} - 16 \frac{2 \sqrt{3}}{3}$

Step 4.1.2.1.5.2

Multiply $8$ by $4$ .

$\frac{32 \sqrt{3}}{9} - 16 \frac{2 \sqrt{3}}{3}$

Step 4.1.2.1.6

Multiply $- 16 \frac{2 \sqrt{3}}{3}$ .

Tap for more steps...

Step 4.1.2.1.6.1

Combine $- 16$ and $\frac{2 \sqrt{3}}{3}$ .

$\frac{32 \sqrt{3}}{9} + \frac{- 16 (2 \sqrt{3})}{3}$

Step 4.1.2.1.6.2

Multiply $2$ by $- 16$ .

$\frac{32 \sqrt{3}}{9} + \frac{- 32 \sqrt{3}}{3}$

Step 4.1.2.1.7

Move the negative in front of the fraction.

$\frac{32 \sqrt{3}}{9} - \frac{32 \sqrt{3}}{3}$

Step 4.1.2.2

To write $- \frac{32 \sqrt{3}}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{32 \sqrt{3}}{9} - \frac{32 \sqrt{3}}{3} \cdot \frac{3}{3}$

Step 4.1.2.3

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

Tap for more steps...

Step 4.1.2.3.1

Multiply $\frac{32 \sqrt{3}}{3}$ by $\frac{3}{3}$ .

$\frac{32 \sqrt{3}}{9} - \frac{32 \sqrt{3} \cdot 3}{3 \cdot 3}$

Step 4.1.2.3.2

Multiply $3$ by $3$ .

$\frac{32 \sqrt{3}}{9} - \frac{32 \sqrt{3} \cdot 3}{9}$

Step 4.1.2.4

Combine the numerators over the common denominator.

$\frac{32 \sqrt{3} - 32 \sqrt{3} \cdot 3}{9}$

Step 4.1.2.5

Simplify the numerator.

Tap for more steps...

Step 4.1.2.5.1

Multiply $3$ by $- 32$ .

$\frac{32 \sqrt{3} - 96 \sqrt{3}}{9}$

Step 4.1.2.5.2

Subtract $96 \sqrt{3}$ from $32 \sqrt{3}$ .

$\frac{- 64 \sqrt{3}}{9}$

Step 4.1.2.6

Move the negative in front of the fraction.

$- \frac{64 \sqrt{3}}{9}$

Step 4.2

Evaluate at $x = - \frac{2 \sqrt{3}}{3}$ .

Tap for more steps...

Step 4.2.1

Substitute $- \frac{2 \sqrt{3}}{3}$ for $x$ .

$4 {(- \frac{2 \sqrt{3}}{3})}^{3} - 16 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2

Simplify.

Tap for more steps...

Step 4.2.2.1

Simplify each term.

Tap for more steps...

Step 4.2.2.1.1

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

Tap for more steps...

Step 4.2.2.1.1.1

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

$4 ({(- 1)}^{3} {(\frac{2 \sqrt{3}}{3})}^{3}) - 16 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2.1.1.2

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

$4 ({(- 1)}^{3} \frac{{(2 \sqrt{3})}^{3}}{3^{3}}) - 16 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2.1.1.3

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

$4 ({(- 1)}^{3} \frac{2^{3} {\sqrt{3}}^{3}}{3^{3}}) - 16 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2.1.2

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

$4 (- \frac{2^{3} {\sqrt{3}}^{3}}{3^{3}}) - 16 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2.1.3

Simplify the numerator.

Tap for more steps...

Step 4.2.2.1.3.1

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

$4 (- \frac{8 {\sqrt{3}}^{3}}{3^{3}}) - 16 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2.1.3.2

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

$4 (- \frac{8 \sqrt{3^{3}}}{3^{3}}) - 16 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2.1.3.3

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

$4 (- \frac{8 \sqrt{27}}{3^{3}}) - 16 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2.1.3.4

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

Tap for more steps...

Step 4.2.2.1.3.4.1

Factor $9$ out of $27$ .

$4 (- \frac{8 \sqrt{9 (3)}}{3^{3}}) - 16 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2.1.3.4.2

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

$4 (- \frac{8 \sqrt{3^{2} \cdot 3}}{3^{3}}) - 16 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2.1.3.5

Pull terms out from under the radical.

$4 (- \frac{8 \cdot 3 \sqrt{3}}{3^{3}}) - 16 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2.1.3.6

Multiply $8$ by $3$ .

$4 (- \frac{24 \sqrt{3}}{3^{3}}) - 16 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2.1.4

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

$4 (- \frac{24 \sqrt{3}}{27}) - 16 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2.1.5

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

Tap for more steps...

Step 4.2.2.1.5.1

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

$4 (- \frac{3 (8 \sqrt{3})}{27}) - 16 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2.1.5.2

Cancel the common factors.

Tap for more steps...

Step 4.2.2.1.5.2.1

Factor $3$ out of $27$ .

$4 (- \frac{3 (8 \sqrt{3})}{3 (9)}) - 16 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2.1.5.2.2

Cancel the common factor.

$4 (- \frac{3 (8 \sqrt{3})}{3 \cdot 9}) - 16 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2.1.5.2.3

Rewrite the expression.

$4 (- \frac{8 \sqrt{3}}{9}) - 16 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2.1.6

Multiply $4 (- \frac{8 \sqrt{3}}{9})$ .

Tap for more steps...

Step 4.2.2.1.6.1

Multiply $- 1$ by $4$ .

$- 4 \frac{8 \sqrt{3}}{9} - 16 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2.1.6.2

Combine $- 4$ and $\frac{8 \sqrt{3}}{9}$ .

$\frac{- 4 (8 \sqrt{3})}{9} - 16 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2.1.6.3

Multiply $8$ by $- 4$ .

$\frac{- 32 \sqrt{3}}{9} - 16 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2.1.7

Move the negative in front of the fraction.

$- \frac{32 \sqrt{3}}{9} - 16 (- \frac{2 \sqrt{3}}{3})$

Step 4.2.2.1.8

Multiply $- 16 (- \frac{2 \sqrt{3}}{3})$ .

Tap for more steps...

Step 4.2.2.1.8.1

Multiply $- 1$ by $- 16$ .

$- \frac{32 \sqrt{3}}{9} + 16 \frac{2 \sqrt{3}}{3}$

Step 4.2.2.1.8.2

Combine $16$ and $\frac{2 \sqrt{3}}{3}$ .

$- \frac{32 \sqrt{3}}{9} + \frac{16 (2 \sqrt{3})}{3}$

Step 4.2.2.1.8.3

Multiply $2$ by $16$ .

$- \frac{32 \sqrt{3}}{9} + \frac{32 \sqrt{3}}{3}$

Step 4.2.2.2

To write $\frac{32 \sqrt{3}}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{32 \sqrt{3}}{9} + \frac{32 \sqrt{3}}{3} \cdot \frac{3}{3}$

Step 4.2.2.3

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

Tap for more steps...

Step 4.2.2.3.1

Multiply $\frac{32 \sqrt{3}}{3}$ by $\frac{3}{3}$ .

$- \frac{32 \sqrt{3}}{9} + \frac{32 \sqrt{3} \cdot 3}{3 \cdot 3}$

Step 4.2.2.3.2

Multiply $3$ by $3$ .

$- \frac{32 \sqrt{3}}{9} + \frac{32 \sqrt{3} \cdot 3}{9}$

Step 4.2.2.4

Combine the numerators over the common denominator.

$\frac{- 32 \sqrt{3} + 32 \sqrt{3} \cdot 3}{9}$

Step 4.2.2.5

Simplify the numerator.

Tap for more steps...

Step 4.2.2.5.1

Multiply $3$ by $32$ .

$\frac{- 32 \sqrt{3} + 96 \sqrt{3}}{9}$

Step 4.2.2.5.2

Add $- 32 \sqrt{3}$ and $96 \sqrt{3}$ .

$\frac{64 \sqrt{3}}{9}$

Step 4.3

List all of the points.

$(\frac{2 \sqrt{3}}{3}, - \frac{64 \sqrt{3}}{9}), (- \frac{2 \sqrt{3}}{3}, \frac{64 \sqrt{3}}{9})$

Step 5