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

$45 = 0$

$x^{2} = 0$

Set the first factor equal to $0$ and solve.

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Set the first factor equal to $0$ .

$45 = 0$

Rewrite the equation as $0 = 45$ .

$0 = 45$

Since $0 \neq 45$ , there are no solutions.

No solution

Set the next factor equal to $0$ and solve.

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Set the next factor equal to $0$ .

$x^{2} = 0$

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, assuming positive real numbers.

$x = \pm 0$

$\pm 0$ 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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Remove parentheses around $0$ .

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

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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