Calculus Examples

$y = 15 x^{3}$

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

$f (x) = 15 x^{3}$

Find the derivative.

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Since $15$ is constant with respect to $x$ , the derivative of $15 x^{3}$ with respect to $x$ is $15 \frac{d}{d x} [x^{3}]$ .

$15 \frac{d}{d x} [x^{3}]$

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

$15 (3 x^{2})$

Multiply $3$ by $15$ .

$45 x^{2}$

Set the derivative equal to $0$ then solve the equation $45 x^{2} = 0$ .

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Divide each term by $45$ and simplify.

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Divide each term in $45 x^{2} = 0$ by $45$ .

$\frac{45 x^{2}}{45} = \frac{0}{45}$

Cancel the common factor of $45$ .

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Cancel the common factor.

$\frac{45 x^{2}}{45} = \frac{0}{45}$

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

$x^{2} = \frac{0}{45}$

Divide $0$ by $45$ .

$x^{2} = 0$

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

$x = \pm \sqrt{0}$

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 $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Pull terms out from under the radical.

$x = \pm | 0 |$

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

$x = \pm 0$

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

$x = 0$

Solve the original function $f (x) = 15 x^{3}$ at $x = 0$ .

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

$f (0) = 15 {(0)}^{3}$

Simplify the result.

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

$f (0) = 15 \cdot 0$

Multiply $15$ by $0$ .

$f (0) = 0$

The final answer is $0$ .

$0$

The horizontal tangent lines on function $f (x) = 15 x^{3}$ are $y = 0$ .

$y = 0$

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