Calculus Examples

$x^{4} - 18 x^{2} + 81$

Step 1

Find the derivative.

Tap for more steps...

Step 1.1

Differentiate.

Tap for more steps...

Step 1.1.1

By the Sum Rule, the derivative of $x^{4} - 18 x^{2} + 81$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 18 x^{2}] + \frac{d}{d x} [81]$ .

$\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 18 x^{2}] + \frac{d}{d x} [81]$

Step 1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$4 x^{3} + \frac{d}{d x} [- 18 x^{2}] + \frac{d}{d x} [81]$

Step 1.2

Evaluate $\frac{d}{d x} [- 18 x^{2}]$ .

Tap for more steps...

Step 1.2.1

Since $- 18$ is constant with respect to $x$ , the derivative of $- 18 x^{2}$ with respect to $x$ is $- 18 \frac{d}{d x} [x^{2}]$ .

$4 x^{3} - 18 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [81]$

Step 1.2.2

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

$4 x^{3} - 18 (2 x) + \frac{d}{d x} [81]$

Step 1.2.3

Multiply $2$ by $- 18$ .

$4 x^{3} - 36 x + \frac{d}{d x} [81]$

Step 1.3

Differentiate using the Constant Rule.

Tap for more steps...

Step 1.3.1

Since $81$ is constant with respect to $x$ , the derivative of $81$ with respect to $x$ is $0$ .

$4 x^{3} - 36 x + 0$

Step 1.3.2

Add $4 x^{3} - 36 x$ and $0$ .

$4 x^{3} - 36 x$

Step 2

Set the derivative equal to $0$ then solve the equation $4 x^{3} - 36 x = 0$ .

Tap for more steps...

Step 2.1

Factor the left side of the equation.

Tap for more steps...

Step 2.1.1

Factor $4 x$ out of $4 x^{3} - 36 x$ .

Tap for more steps...

Step 2.1.1.1

Factor $4 x$ out of $4 x^{3}$ .

$4 x (x^{2}) - 36 x = 0$

Step 2.1.1.2

Factor $4 x$ out of $- 36 x$ .

$4 x (x^{2}) + 4 x (- 9) = 0$

Step 2.1.1.3

Factor $4 x$ out of $4 x (x^{2}) + 4 x (- 9)$ .

$4 x (x^{2} - 9) = 0$

Step 2.1.2

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

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

Step 2.1.3

Factor.

Tap for more steps...

Step 2.1.3.1

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 $b = 3$ .

$4 x ((x + 3) (x - 3)) = 0$

Step 2.1.3.2

Remove unnecessary parentheses.

$4 x (x + 3) (x - 3) = 0$

Step 2.2

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

$x = 0$

$x + 3 = 0$

$x - 3 = 0$

Step 2.3

Set $x$ equal to $0$ .

$x = 0$

Step 2.4

Set $x + 3$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.4.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 2.4.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 2.5

Set $x - 3$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.5.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 2.5.2

Add $3$ to both sides of the equation.

$x = 3$

Step 2.6

The final solution is all the values that make $4 x (x + 3) (x - 3) = 0$ true.

$x = 0, - 3, 3$

Step 3

Solve the original function $f (x) = x^{4} - 18 x^{2} + 81$ at $x = 0$ .

Tap for more steps...

Step 3.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = {(0)}^{4} - 18 {(0)}^{2} + 81$

Step 3.2

Simplify the result.

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

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

$f (0) = 0 - 18 {(0)}^{2} + 81$

Step 3.2.1.2

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

$f (0) = 0 - 18 \cdot 0 + 81$

Step 3.2.1.3

Multiply $- 18$ by $0$ .

$f (0) = 0 + 0 + 81$

Step 3.2.2

Simplify by adding numbers.

Tap for more steps...

Step 3.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 81$

Step 3.2.2.2

Add $0$ and $81$ .

$f (0) = 81$

Step 3.2.3

The final answer is $81$ .

$81$

Step 4

Solve the original function $f (x) = x^{4} - 18 x^{2} + 81$ at $x = - 3$ .

Tap for more steps...

Step 4.1

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = {(- 3)}^{4} - 18 {(- 3)}^{2} + 81$

Step 4.2

Simplify the result.

Tap for more steps...

Step 4.2.1

Simplify each term.

Tap for more steps...

Step 4.2.1.1

Raise $- 3$ to the power of $4$ .

$f (- 3) = 81 - 18 {(- 3)}^{2} + 81$

Step 4.2.1.2

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

$f (- 3) = 81 - 18 \cdot 9 + 81$

Step 4.2.1.3

Multiply $- 18$ by $9$ .

$f (- 3) = 81 - 162 + 81$

Step 4.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 4.2.2.1

Subtract $162$ from $81$ .

$f (- 3) = - 81 + 81$

Step 4.2.2.2

Add $- 81$ and $81$ .

$f (- 3) = 0$

Step 4.2.3

The final answer is $0$ .

$0$

Step 5

The horizontal tangent lines on function $f (x) = x^{4} - 18 x^{2} + 81$ are $y = 81, y = 0, y = 0$ .

$y = 81, y = 0, y = 0$

Step 6