Calculus Examples

$f (x) = x^{2} + 6 x$

Step 1

Find the derivative.

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Step 1.1

Differentiate.

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Step 1.1.1

By the Sum Rule, the derivative of $x^{2} + 6 x$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [6 x]$ .

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

Step 1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

$2 x + 6 \cdot 1$

Step 1.2.3

Multiply $6$ by $1$ .

$2 x + 6$

Step 2

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

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Step 2.1

Subtract $6$ from both sides of the equation.

$2 x = - 6$

Step 2.2

Divide each term in $2 x = - 6$ by $2$ and simplify.

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Step 2.2.1

Divide each term in $2 x = - 6$ by $2$ .

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

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.1.1

Cancel the common factor.

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

Step 2.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $- 6$ by $2$ .

$x = - 3$

Step 3

Solve the original function $f (x) = x^{2} + 6 x$ at $x = - 3$ .

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Step 3.1

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = {(- 3)}^{2} + 6 (- 3)$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Raise $- 3$ to the power of $2$ .

$f (- 3) = 9 + 6 (- 3)$

Step 3.2.1.2

Multiply $6$ by $- 3$ .

$f (- 3) = 9 - 18$

Step 3.2.2

Subtract $18$ from $9$ .

$f (- 3) = - 9$

Step 3.2.3

The final answer is $- 9$ .

$- 9$

Step 4

The horizontal tangent line on function $f (x) = x^{2} + 6 x$ is $y = - 9$ .

$y = - 9$

Step 5