Calculus Examples

$y = x^{3}$

Step 1

Find the first derivative.

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

$f' (x) = 3 x^{2}$

The first derivative of $f (x)$ with respect to $x$ is $3 x^{2}$ .

$3 x^{2}$

Step 2

Set the first derivative equal to $0$ then solve the equation $3 x^{2} = 0$ .

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Set the first derivative equal to $0$ .

$3 x^{2} = 0$

Divide each term in $3 x^{2} = 0$ by $3$ and simplify.

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Divide each term in $3 x^{2} = 0$ by $3$ .

$\frac{3 x^{2}}{3} = \frac{0}{3}$

Simplify the left side.

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Cancel the common factor of $3$ .

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Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{0}{3}$

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

$x^{2} = \frac{0}{3}$

Simplify the right side.

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Divide $0$ by $3$ .

$x^{2} = 0$

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

$x = \pm \sqrt{0}$

Simplify $\pm \sqrt{0}$ .

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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.

$x = \pm 0$

Plus or minus $0$ is $0$ .

$x = 0$

Step 3

Find the values where the derivative is undefined.

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The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

Evaluate $x^{3}$ at each $x$ value where the derivative is $0$ or undefined.

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Evaluate at $x = 0$ .

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Substitute $0$ for $x$ .

${(0)}^{3}$

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

$0$

List all of the points.

$(0, 0)$

Step 5