Calculus Examples

$y = x^{9}$

Step 1

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

$f (x) = x^{9}$

Step 2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 9$ .

$9 x^{8}$

Step 3

Set the derivative equal to $0$ then solve the equation $9 x^{8} = 0$ .

Tap for more steps...

Step 3.1

Divide each term in $9 x^{8} = 0$ by $9$ and simplify.

Tap for more steps...

Step 3.1.1

Divide each term in $9 x^{8} = 0$ by $9$ .

$\frac{9 x^{8}}{9} = \frac{0}{9}$

Step 3.1.2

Simplify the left side.

Tap for more steps...

Step 3.1.2.1

Cancel the common factor of $9$ .

Tap for more steps...

Step 3.1.2.1.1

Cancel the common factor.

$\frac{9 x^{8}}{9} = \frac{0}{9}$

Step 3.1.2.1.2

Divide $x^{8}$ by $1$ .

$x^{8} = \frac{0}{9}$

Step 3.1.3

Simplify the right side.

Tap for more steps...

Step 3.1.3.1

Divide $0$ by $9$ .

$x^{8} = 0$

Step 3.2

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

$x = \pm \sqrt[8]{0}$

Step 3.3

Simplify $\pm \sqrt[8]{0}$ .

Tap for more steps...

Step 3.3.1

Rewrite $0$ as $0^{8}$ .

$x = \pm \sqrt[8]{0^{8}}$

Step 3.3.2

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

$x = \pm 0$

Step 3.3.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 4

Solve the original function $f (x) = x^{9}$ at $x = 0$ .

Tap for more steps...

Step 4.1

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

$f (0) = {(0)}^{9}$

Step 4.2

Simplify the result.

Tap for more steps...

Step 4.2.1

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

$f (0) = 0$

Step 4.2.2

The final answer is $0$ .

$0$

Step 5

The horizontal tangent line on function $f (x) = x^{9}$ is $y = 0$ .

$y = 0$

Step 6

Enter YOUR Problem