Calculus Examples

$y = x^{3} + 6 x$

Step 1

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

$f (x) = x^{3} + 6 x$

Step 2

Find the derivative.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $x^{3} + 6 x$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [6 x]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [6 x]$

Step 2.1.2

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

$3 x^{2} + \frac{d}{d x} [6 x]$

Step 2.2

Evaluate $\frac{d}{d x} [6 x]$ .

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Step 2.2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$3 x^{2} + 6 \frac{d}{d x} [x]$

Step 2.2.2

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

$3 x^{2} + 6 \cdot 1$

Step 2.2.3

Multiply $6$ by $1$ .

$3 x^{2} + 6$

Step 3

Set the derivative equal to $0$ then solve the equation $3 x^{2} + 6 = 0$ .

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Step 3.1

Subtract $6$ from both sides of the equation.

$3 x^{2} = - 6$

Step 3.2

Divide each term in $3 x^{2} = - 6$ by $3$ and simplify.

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Step 3.2.1

Divide each term in $3 x^{2} = - 6$ by $3$ .

$\frac{3 x^{2}}{3} = \frac{- 6}{3}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $3$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{- 6}{3}$

Step 3.2.2.1.2

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

$x^{2} = \frac{- 6}{3}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $- 6$ by $3$ .

$x^{2} = - 2$

Step 3.3

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

$x = \pm \sqrt{- 2}$

Step 3.4

Simplify $\pm \sqrt{- 2}$ .

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Step 3.4.1

Rewrite $- 2$ as $- 1 (2)$ .

$x = \pm \sqrt{- 1 (2)}$

Step 3.4.2

Rewrite $\sqrt{- 1 (2)}$ as $\sqrt{- 1} \cdot \sqrt{2}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{2}$

Step 3.4.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \sqrt{2}$

Step 3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = i \sqrt{2}$

Step 3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - i \sqrt{2}$

Step 3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = i \sqrt{2}, - i \sqrt{2}$

Step 4

A tangent line cannot be found at an imaginary point. The point at $x =$ does not exist on the real coordinate system.

A tangent cannot be found from the root

Step 5

There are no horizontal tangent lines on the function $f (x) = x^{3} + 6 x$ .

No horizontal tangent lines

Step 6