Calculus Examples

$y = x^{9}$

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

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

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

$9 x^{8}$

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

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Rewrite $9 x^{8}$ as a set of linear factors.

$9 (x^{8}) = 0$

Divide each term by $9$ and simplify.

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

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

Simplify the left side of the equation by cancelling the common factors.

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Multiply $9$ by $x^{8}$ to get $9 x^{8}$ .

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

Reduce the expression by cancelling the common factors.

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Cancel the common factor.

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

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

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

Divide $0$ by $9$ to get $0$ .

$x^{8} = 0$

Take the $8 t h$ root of both sides of the equation to eliminate the exponent on the left side.

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

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

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Simplify the right side of the equation.

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

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

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

$x = \pm 0$

$\pm 0$ is equal to $0$ .

$x = 0$

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

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

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

Simplify the result.

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Remove parentheses around $0$ .

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

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

$f (0) = 0$

The final answer is $0$ .

$0$

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

$y = 0$

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Calculus Examples

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