Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

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

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

Step 1.2

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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Step 2.1

Set the first derivative equal to $0$ .

$3 x^{2} = 0$

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $3$ .

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Step 2.2.2.1.1

Cancel the common factor.

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

Step 2.2.2.1.2

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

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

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $0$ by $3$ .

$x^{2} = 0$

Step 2.3

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

$x = \pm \sqrt{0}$

Step 2.4

Simplify $\pm \sqrt{0}$ .

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Step 2.4.1

Rewrite $0$ as $0^{2}$ .

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

Step 2.4.2

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

$x = \pm 0$

Step 2.4.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

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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Step 4.1

Evaluate at $x = 0$ .

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Step 4.1.1

Substitute $0$ for $x$ .

${(0)}^{3}$

Step 4.1.2

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

$0$

Step 4.2

List all of the points.

$(0, 0)$

Step 5

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