Calculus Examples

$f (x) = x^{3} + a x$

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

Differentiate.

Tap for more steps...

Step 1.1.1.1

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

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

Step 1.1.1.2

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

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

Step 1.1.2

Evaluate $\frac{d}{d x} [a x]$ .

Tap for more steps...

Step 1.1.2.1

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

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

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 $n = 1$ .

$3 x^{2} + a \cdot 1$

Step 1.1.2.3

Multiply $a$ by $1$ .

$f' (x) = 3 x^{2} + a$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $3 x^{2} + a$ .

$3 x^{2} + a$

Step 2

Set the first derivative equal to $0$ then solve the equation $3 x^{2} + a = 0$ .

Tap for more steps...

Step 2.1

Set the first derivative equal to $0$ .

$3 x^{2} + a = 0$

Step 2.2

Subtract $a$ from both sides of the equation.

$3 x^{2} = - a$

Step 2.3

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

Tap for more steps...

Step 2.3.1

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

$\frac{3 x^{2}}{3} = \frac{- a}{3}$

Step 2.3.2

Simplify the left side.

Tap for more steps...

Step 2.3.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.3.2.1.1

Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{- a}{3}$

Step 2.3.2.1.2

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

$x^{2} = \frac{- a}{3}$

Step 2.3.3

Simplify the right side.

Tap for more steps...

Step 2.3.3.1

Move the negative in front of the fraction.

$x^{2} = - \frac{a}{3}$

Step 2.4

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

$x = \pm \sqrt{- \frac{a}{3}}$

Step 2.5

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

Tap for more steps...

Step 2.5.1

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

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

Step 2.5.2

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

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

Step 2.5.3

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

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

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

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

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

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

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

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 $x^{3} + a x$ at each $x$ value where the derivative is $0$ or undefined.

Tap for more steps...

Step 4.1

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

Tap for more steps...

Step 4.1.1

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

${(\sqrt{- \frac{a}{3}})}^{3} + a (\sqrt{- \frac{a}{3}})$

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

Rewrite ${\sqrt{- \frac{a}{3}}}^{3}$ as $\sqrt{{(- \frac{a}{3})}^{3}}$ .

$\sqrt{{(- \frac{a}{3})}^{3}} + a (\sqrt{- \frac{a}{3}})$

Step 4.1.2.1.2

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

$\sqrt{{(- 1)}^{3} {(\frac{a}{3})}^{3}} + a (\sqrt{- \frac{a}{3}})$

Step 4.1.2.1.3

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

$\sqrt{- {(\frac{a}{3})}^{3}} + a (\sqrt{- \frac{a}{3}})$

Step 4.1.2.1.4

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

$\sqrt{- \frac{a^{3}}{3^{3}}} + a (\sqrt{- \frac{a}{3}})$

Step 4.1.2.1.5

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

$\sqrt{- \frac{a^{3}}{27}} + a (\sqrt{- \frac{a}{3}})$

Step 4.1.2.1.6

Rewrite $- \frac{a^{3}}{27}$ as ${(\frac{a}{3})}^{2} \cdot (- \frac{a}{3})$ .

Tap for more steps...

Step 4.1.2.1.6.1

Factor the perfect power $a^{2}$ out of $a^{3}$ .

$\sqrt{- \frac{a^{2} a}{27}} + a (\sqrt{- \frac{a}{3}})$

Step 4.1.2.1.6.2

Factor the perfect power $3^{2}$ out of $27$ .

$\sqrt{- \frac{a^{2} a}{3^{2} \cdot 3}} + a (\sqrt{- \frac{a}{3}})$

Step 4.1.2.1.6.3

Rearrange the fraction $\frac{a^{2} a}{3^{2} \cdot 3}$ .

$\sqrt{- ({(\frac{a}{3})}^{2} \frac{a}{3})} + a (\sqrt{- \frac{a}{3}})$

Step 4.1.2.1.6.4

Reorder $- 1$ and ${(\frac{a}{3})}^{2}$ .

$\sqrt{{(\frac{a}{3})}^{2} \cdot - 1 \frac{a}{3}} + a (\sqrt{- \frac{a}{3}})$

Step 4.1.2.1.6.5

Add parentheses.

$\sqrt{{(\frac{a}{3})}^{2} \cdot (- \frac{a}{3})} + a (\sqrt{- \frac{a}{3}})$

Step 4.1.2.1.7

Pull terms out from under the radical.

$\frac{a}{3} \sqrt{- \frac{a}{3}} + a (\sqrt{- \frac{a}{3}})$

Step 4.1.2.1.8

Combine $\frac{a}{3}$ and $\sqrt{- \frac{a}{3}}$ .

$\frac{a \sqrt{- \frac{a}{3}}}{3} + a (\sqrt{- \frac{a}{3}})$

Step 4.1.2.2

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

$\frac{a \sqrt{- \frac{a}{3}}}{3} + a (\sqrt{- \frac{a}{3}}) \cdot \frac{3}{3}$

Step 4.1.2.3

Simplify terms.

Tap for more steps...

Step 4.1.2.3.1

Combine $a (\sqrt{- \frac{a}{3}})$ and $\frac{3}{3}$ .

$\frac{a \sqrt{- \frac{a}{3}}}{3} + \frac{a (\sqrt{- \frac{a}{3}}) \cdot 3}{3}$

Step 4.1.2.3.2

Combine the numerators over the common denominator.

$\frac{a \sqrt{- \frac{a}{3}} + a (\sqrt{- \frac{a}{3}}) \cdot 3}{3}$

Step 4.1.2.4

Simplify the numerator.

Tap for more steps...

Step 4.1.2.4.1

Move $3$ to the left of $a (\sqrt{- \frac{a}{3}})$ .

$\frac{a \sqrt{- \frac{a}{3}} + 3 \cdot (a (\sqrt{- \frac{a}{3}}))}{3}$

Step 4.1.2.4.2

Add $a \sqrt{- \frac{a}{3}}$ and $3 a \sqrt{- \frac{a}{3}}$ .

$\frac{4 a \sqrt{- \frac{a}{3}}}{3}$

Step 4.2

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

Tap for more steps...

Step 4.2.1

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

${(- \sqrt{- \frac{a}{3}})}^{3} + a (- \sqrt{- \frac{a}{3}})$

Step 4.2.2

Simplify each term.

Tap for more steps...

Step 4.2.2.1

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

${(- 1)}^{3} {\sqrt{- \frac{a}{3}}}^{3} + a (- \sqrt{- \frac{a}{3}})$

Step 4.2.2.2

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

$- {\sqrt{- \frac{a}{3}}}^{3} + a (- \sqrt{- \frac{a}{3}})$

Step 4.2.2.3

Rewrite ${\sqrt{- \frac{a}{3}}}^{3}$ as $\sqrt{{(- \frac{a}{3})}^{3}}$ .

$- \sqrt{{(- \frac{a}{3})}^{3}} + a (- \sqrt{- \frac{a}{3}})$

Step 4.2.2.4

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

$- \sqrt{{(- 1)}^{3} {(\frac{a}{3})}^{3}} + a (- \sqrt{- \frac{a}{3}})$

Step 4.2.2.5

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

$- \sqrt{- {(\frac{a}{3})}^{3}} + a (- \sqrt{- \frac{a}{3}})$

Step 4.2.2.6

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

$- \sqrt{- \frac{a^{3}}{3^{3}}} + a (- \sqrt{- \frac{a}{3}})$

Step 4.2.2.7

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

$- \sqrt{- \frac{a^{3}}{27}} + a (- \sqrt{- \frac{a}{3}})$

Step 4.2.2.8

Rewrite $- \frac{a^{3}}{27}$ as ${(\frac{a}{3})}^{2} \cdot (- \frac{a}{3})$ .

Tap for more steps...

Step 4.2.2.8.1

Factor the perfect power $a^{2}$ out of $a^{3}$ .

$- \sqrt{- \frac{a^{2} a}{27}} + a (- \sqrt{- \frac{a}{3}})$

Step 4.2.2.8.2

Factor the perfect power $3^{2}$ out of $27$ .

$- \sqrt{- \frac{a^{2} a}{3^{2} \cdot 3}} + a (- \sqrt{- \frac{a}{3}})$

Step 4.2.2.8.3

Rearrange the fraction $\frac{a^{2} a}{3^{2} \cdot 3}$ .

$- \sqrt{- ({(\frac{a}{3})}^{2} \frac{a}{3})} + a (- \sqrt{- \frac{a}{3}})$

Step 4.2.2.8.4

Reorder $- 1$ and ${(\frac{a}{3})}^{2}$ .

$- \sqrt{{(\frac{a}{3})}^{2} \cdot - 1 \frac{a}{3}} + a (- \sqrt{- \frac{a}{3}})$

Step 4.2.2.8.5

Add parentheses.

$- \sqrt{{(\frac{a}{3})}^{2} \cdot (- \frac{a}{3})} + a (- \sqrt{- \frac{a}{3}})$

Step 4.2.2.9

Pull terms out from under the radical.

$- (\frac{a}{3} \sqrt{- \frac{a}{3}}) + a (- \sqrt{- \frac{a}{3}})$

Step 4.2.2.10

Combine $\frac{a}{3}$ and $\sqrt{- \frac{a}{3}}$ .

$- \frac{a \sqrt{- \frac{a}{3}}}{3} + a (- \sqrt{- \frac{a}{3}})$

Step 4.2.2.11

Rewrite using the commutative property of multiplication.

$- \frac{a \sqrt{- \frac{a}{3}}}{3} - a \sqrt{- \frac{a}{3}}$

Step 4.3

List all of the points.

$(\sqrt{- \frac{a}{3}}, \frac{4 a \sqrt{- \frac{a}{3}}}{3}), (- \sqrt{- \frac{a}{3}}, - \frac{a \sqrt{- \frac{a}{3}}}{3} - a \sqrt{- \frac{a}{3}})$

Step 5