Calculus Examples

$y = x^{3} + 3$

Step 1

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

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

Step 2

Find the derivative.

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Step 2.1

By the Sum Rule, the derivative of $x^{3} + 3$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [3]$ .

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

Step 2.2

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

$3 x^{2} + \frac{d}{d x} [3]$

Step 2.3

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$3 x^{2} + 0$

Step 2.4

Add $3 x^{2}$ and $0$ .

$3 x^{2}$

Step 3

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

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $3$ .

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Step 3.1.2.1.1

Cancel the common factor.

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

Step 3.1.2.1.2

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

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

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Divide $0$ by $3$ .

$x^{2} = 0$

Step 3.2

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

$x = \pm \sqrt{0}$

Step 3.3

Simplify $\pm \sqrt{0}$ .

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Step 3.3.1

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

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

Step 3.3.2

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

$x = \pm 0$

Step 3.3.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 4

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

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Step 4.1

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

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

Step 4.2

Simplify the result.

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Step 4.2.1

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

$f (0) = 0 + 3$

Step 4.2.2

Add $0$ and $3$ .

$f (0) = 3$

Step 4.2.3

The final answer is $3$ .

$3$

Step 5

The horizontal tangent line on function $f (x) = x^{3} + 3$ is $y = 3$ .

$y = 3$

Step 6