Calculus Examples

$f (x) = \frac{x + 6}{x + 2}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x + 6$ and $g (x) = x + 2$ .

$f' (x) = \frac{(x + 2) \frac{d}{d x} (x + 6) - (x + 6) \frac{d}{d x} (x + 2)}{{(x + 2)}^{2}}$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

$f' (x) = \frac{(x + 2) (\frac{d}{d x} (x) + \frac{d}{d x} (6)) - (x + 6) \frac{d}{d x} (x + 2)}{{(x + 2)}^{2}}$

Step 1.1.2.2

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

$f' (x) = \frac{(x + 2) (1 + \frac{d}{d x} (6)) - (x + 6) \frac{d}{d x} (x + 2)}{{(x + 2)}^{2}}$

Step 1.1.2.3

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$f' (x) = \frac{(x + 2) (1 + 0) - (x + 6) \frac{d}{d x} (x + 2)}{{(x + 2)}^{2}}$

Step 1.1.2.4

Simplify the expression.

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Step 1.1.2.4.1

Add $1$ and $0$ .

$f' (x) = \frac{(x + 2) \cdot 1 - (x + 6) \frac{d}{d x} (x + 2)}{{(x + 2)}^{2}}$

Step 1.1.2.4.2

Multiply $x + 2$ by $1$ .

$f' (x) = \frac{x + 2 - (x + 6) \frac{d}{d x} (x + 2)}{{(x + 2)}^{2}}$

Step 1.1.2.5

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

$f' (x) = \frac{x + 2 - (x + 6) (\frac{d}{d x} (x) + \frac{d}{d x} (2))}{{(x + 2)}^{2}}$

Step 1.1.2.6

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

$f' (x) = \frac{x + 2 - (x + 6) (1 + \frac{d}{d x} (2))}{{(x + 2)}^{2}}$

Step 1.1.2.7

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$f' (x) = \frac{x + 2 - (x + 6) (1 + 0)}{{(x + 2)}^{2}}$

Step 1.1.2.8

Simplify the expression.

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Step 1.1.2.8.1

Add $1$ and $0$ .

$f' (x) = \frac{x + 2 - (x + 6) \cdot 1}{{(x + 2)}^{2}}$

Step 1.1.2.8.2

Multiply $- 1$ by $1$ .

$f' (x) = \frac{x + 2 - (x + 6)}{{(x + 2)}^{2}}$

Step 1.1.3

Simplify.

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Step 1.1.3.1

Apply the distributive property.

$f' (x) = \frac{x + 2 - x - 1 \cdot 6}{{(x + 2)}^{2}}$

Step 1.1.3.2

Simplify the numerator.

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Step 1.1.3.2.1

Combine the opposite terms in $x + 2 - x - 1 \cdot 6$ .

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Step 1.1.3.2.1.1

Subtract $x$ from $x$ .

$f' (x) = \frac{0 + 2 - 1 \cdot 6}{{(x + 2)}^{2}}$

Step 1.1.3.2.1.2

Add $0$ and $2$ .

$f' (x) = \frac{2 - 1 \cdot 6}{{(x + 2)}^{2}}$

Step 1.1.3.2.2

Multiply $- 1$ by $6$ .

$f' (x) = \frac{2 - 6}{{(x + 2)}^{2}}$

Step 1.1.3.2.3

Subtract $6$ from $2$ .

$f' (x) = \frac{- 4}{{(x + 2)}^{2}}$

Step 1.1.3.3

Move the negative in front of the fraction.

$f' (x) = - \frac{4}{{(x + 2)}^{2}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{4}{{(x + 2)}^{2}}$ .

$- \frac{4}{{(x + 2)}^{2}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{4}{{(x + 2)}^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- \frac{4}{{(x + 2)}^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

$4 = 0$

Step 2.3

Since $4 \neq 0$ , there are no solutions.

No solution

Step 3

There are no values of $x$ in the domain of the original problem where the derivative is $0$ or undefined.

No critical points found

Step 4

Find where the derivative is undefined.

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Step 4.1

Set the denominator in $\frac{4}{{(x + 2)}^{2}}$ equal to $0$ to find where the expression is undefined.

${(x + 2)}^{2} = 0$

Step 4.2

Solve for $x$ .

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Step 4.2.1

Set the $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 4.2.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 5

After finding the point that makes the derivative $f' (x) = - \frac{4}{{(x + 2)}^{2}}$ equal to $0$ or undefined, the interval to check where $f (x) = \frac{x + 6}{x + 2}$ is increasing and where it is decreasing is $(- \infty, - 2) \cup (- 2, \infty)$ .

$(- \infty, - 2) \cup (- 2, \infty)$

Step 6

Substitute a value from the interval $(- \infty, - 2)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

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

$f' (- 3) = - \frac{4}{{((- 3) + 2)}^{2}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the denominator.

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Step 6.2.1.1

Add $- 3$ and $2$ .

$f' (- 3) = - \frac{4}{{(- 1)}^{2}}$

Step 6.2.1.2

Raise $- 1$ to the power of $2$ .

$f' (- 3) = - \frac{4}{1}$

Step 6.2.2

Simplify the expression.

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Step 6.2.2.1

Divide $4$ by $1$ .

$f' (- 3) = - 1 \cdot 4$

Step 6.2.2.2

Multiply $- 1$ by $4$ .

$f' (- 3) = - 4$

Step 6.2.3

The final answer is $- 4$ .

$- 4$

Step 6.3

At $x = - 3$ the derivative is $- 4$ . Since this is negative, the function is decreasing on $(- \infty, - 2)$ .

Decreasing on $(- \infty, - 2)$ since $f' (x) < 0$

Step 7

Substitute a value from the interval $(- 2, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $- 1$ in the expression.

$f' (- 1) = - \frac{4}{{((- 1) + 2)}^{2}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify the denominator.

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Step 7.2.1.1

Add $- 1$ and $2$ .

$f' (- 1) = - \frac{4}{1^{2}}$

Step 7.2.1.2

One to any power is one.

$f' (- 1) = - \frac{4}{1}$

Step 7.2.2

Simplify the expression.

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Step 7.2.2.1

Divide $4$ by $1$ .

$f' (- 1) = - 1 \cdot 4$

Step 7.2.2.2

Multiply $- 1$ by $4$ .

$f' (- 1) = - 4$

Step 7.2.3

The final answer is $- 4$ .

$- 4$

Step 7.3

At $x = - 1$ the derivative is $- 4$ . Since this is negative, the function is decreasing on $(- 2, \infty)$ .

Decreasing on $(- 2, \infty)$ since $f' (x) < 0$

Step 8

List the intervals on which the function is increasing and decreasing.

Decreasing on: $(- \infty, - 2), (- 2, \infty)$

Step 9