Calculus Examples

y = x^{9}

$y = x^{9}$

Step 1

Set

y

$y$ as a function of

x

$x$ .

f (x) = x^{9}

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

Step 2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 9

$n = 9$ .

9 x^{8}

$9 x^{8}$

Step 3

Set the derivative equal to

0

$0$ then solve the equation

9 x^{8} = 0

$9 x^{8} = 0$ .

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Step 3.1

Divide each term in

9 x^{8} = 0

$9 x^{8} = 0$ by

9

$9$ and simplify.

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Step 3.1.1

Divide each term in

9 x^{8} = 0

$9 x^{8} = 0$ by

9

$9$ .

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

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

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of

9

$9$ .

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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.

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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

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

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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

$x = 0$ .

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Step 4.1

Replace the variable

$x$ with

$0$ in the expression.

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

Step 4.2

Simplify the result.

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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

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