Calculus Examples

$x + \frac{6}{x}$

Step 1

Write $x + \frac{6}{x}$ as a function.

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

Step 2

Find the first derivative.

Tap for more steps...

Step 2.1

Find the first derivative.

Tap for more steps...

Step 2.1.1

Differentiate.

Tap for more steps...

Step 2.1.1.1

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

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

Step 2.1.1.2

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

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

Step 2.1.2

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

Tap for more steps...

Step 2.1.2.1

Since $6$ is constant with respect to $x$ , the derivative of $\frac{6}{x}$ with respect to $x$ is $6 \frac{d}{d x} [\frac{1}{x}]$ .

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

Step 2.1.2.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

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

Step 2.1.2.3

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

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

Step 2.1.2.4

Multiply $- 1$ by $6$ .

$f' (x) = 1 - 6 x^{- 2}$

Step 2.1.3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.1.4

Simplify.

Tap for more steps...

Step 2.1.4.1

Combine terms.

Tap for more steps...

Step 2.1.4.1.1

Combine $- 6$ and $\frac{1}{x^{2}}$ .

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

Step 2.1.4.1.2

Move the negative in front of the fraction.

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

Step 2.1.4.2

Reorder terms.

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

Step 2.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{6}{x^{2}} + 1$ .

$- \frac{6}{x^{2}} + 1$

Step 3

Set the first derivative equal to $0$ then solve the equation $- \frac{6}{x^{2}} + 1 = 0$ .

Tap for more steps...

Step 3.1

Set the first derivative equal to $0$ .

$- \frac{6}{x^{2}} + 1 = 0$

Step 3.2

Subtract $1$ from both sides of the equation.

$- \frac{6}{x^{2}} = - 1$

Step 3.3

Find the LCD of the terms in the equation.

Tap for more steps...

Step 3.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x^{2}, 1$

Step 3.3.2

The LCM of one and any expression is the expression.

$x^{2}$

Step 3.4

Multiply each term in $- \frac{6}{x^{2}} = - 1$ by $x^{2}$ to eliminate the fractions.

Tap for more steps...

Step 3.4.1

Multiply each term in $- \frac{6}{x^{2}} = - 1$ by $x^{2}$ .

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

Step 3.4.2

Simplify the left side.

Tap for more steps...

Step 3.4.2.1

Cancel the common factor of $x^{2}$ .

Tap for more steps...

Step 3.4.2.1.1

Move the leading negative in $- \frac{6}{x^{2}}$ into the numerator.

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

Step 3.4.2.1.2

Cancel the common factor.

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

Step 3.4.2.1.3

Rewrite the expression.

$- 6 = - x^{2}$

Step 3.5

Solve the equation.

Tap for more steps...

Step 3.5.1

Rewrite the equation as $- x^{2} = - 6$ .

$- x^{2} = - 6$

Step 3.5.2

Divide each term in $- x^{2} = - 6$ by $- 1$ and simplify.

Tap for more steps...

Step 3.5.2.1

Divide each term in $- x^{2} = - 6$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 6}{- 1}$

Step 3.5.2.2

Simplify the left side.

Tap for more steps...

Step 3.5.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 6}{- 1}$

Step 3.5.2.2.2

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

$x^{2} = \frac{- 6}{- 1}$

Step 3.5.2.3

Simplify the right side.

Tap for more steps...

Step 3.5.2.3.1

Divide $- 6$ by $- 1$ .

$x^{2} = 6$

Step 3.5.3

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

$x = \pm \sqrt{6}$

Step 3.5.4

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

Tap for more steps...

Step 3.5.4.1

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

$x = \sqrt{6}$

Step 3.5.4.2

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

$x = - \sqrt{6}$

Step 3.5.4.3

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

$x = \sqrt{6}, - \sqrt{6}$

Step 4

The values which make the derivative equal to $0$ are $\sqrt{6}, - \sqrt{6}$ .

$\sqrt{6}, - \sqrt{6}$

Step 5

Find where the derivative is undefined.

Tap for more steps...

Step 5.1

Set the denominator in $\frac{6}{x^{2}}$ equal to $0$ to find where the expression is undefined.

$x^{2} = 0$

Step 5.2

Solve for $x$ .

Tap for more steps...

Step 5.2.1

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

$x = \pm \sqrt{0}$

Step 5.2.2

Simplify $\pm \sqrt{0}$ .

Tap for more steps...

Step 5.2.2.1

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

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

Step 5.2.2.2

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

$x = \pm 0$

Step 5.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 6

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the derivative $0$ or undefined.

$(- \infty, - \sqrt{6}) \cup (- \sqrt{6}, 0) \cup (0, \sqrt{6}) \cup (\sqrt{6}, \infty)$

Step 7

Substitute a value from the interval $(- \infty, - \sqrt{6})$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 7.1

Replace the variable $x$ with $- 3.4494898$ in the expression.

$f' (- 3.4494898) = - \frac{6}{{(- 3.4494898)}^{2}} + 1$

Step 7.2

Simplify the result.

Tap for more steps...

Step 7.2.1

Simplify each term.

Tap for more steps...

Step 7.2.1.1

Raise $- 3.4494898$ to the power of $2$ .

$f' (- 3.4494898) = - \frac{6}{11.89897988} + 1$

Step 7.2.1.2

Divide $6$ by $11.89897988$ .

$f' (- 3.4494898) = - 1 \cdot 0.5042449 + 1$

Step 7.2.1.3

Multiply $- 1$ by $0.5042449$ .

$f' (- 3.4494898) = - 0.5042449 + 1$

Step 7.2.2

Add $- 0.5042449$ and $1$ .

$f' (- 3.4494898) = 0.49575509$

Step 7.2.3

The final answer is $0.49575509$ .

$0.49575509$

Step 7.3

At $x = - 3.4494898$ the derivative is $0.49575509$ . Since this is positive, the function is increasing on $(- \infty, - \sqrt{6})$ .

Increasing on $(- \infty, - \sqrt{6})$ since $f' (x) > 0$

Step 8

Substitute a value from the interval $(- 2.4494898, 0)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 8.1

Replace the variable $x$ with $- 1.2247449$ in the expression.

$f' (- 1.2247449) = - \frac{6}{{(- 1.2247449)}^{2}} + 1$

Step 8.2

Simplify the result.

Tap for more steps...

Step 8.2.1

Simplify each term.

Tap for more steps...

Step 8.2.1.1

Raise $- 1.2247449$ to the power of $2$ .

$f' (- 1.2247449) = - \frac{6}{1.50000007} + 1$

Step 8.2.1.2

Divide $6$ by $1.50000007$ .

$f' (- 1.2247449) = - 1 \cdot 3.99999981 + 1$

Step 8.2.1.3

Multiply $- 1$ by $3.99999981$ .

$f' (- 1.2247449) = - 3.99999981 + 1$

Step 8.2.2

Add $- 3.99999981$ and $1$ .

$f' (- 1.2247449) = - 2.99999981$

Step 8.2.3

The final answer is $- 2.99999981$ .

$- 2.99999981$

Step 8.3

At $x = - 1.2247449$ the derivative is $- 2.99999981$ . Since this is negative, the function is decreasing on $(- 2.4494898, 0)$ .

Decreasing on $(- \sqrt{6}, 0)$ since $f' (x) < 0$

Step 9

Substitute a value from the interval $(0, \sqrt{6})$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 9.1

Replace the variable $x$ with $1.2247449$ in the expression.

$f' (1.2247449) = - \frac{6}{{(1.2247449)}^{2}} + 1$

Step 9.2

Simplify the result.

Tap for more steps...

Step 9.2.1

Simplify each term.

Tap for more steps...

Step 9.2.1.1

Raise $1.2247449$ to the power of $2$ .

$f' (1.2247449) = - \frac{6}{1.50000007} + 1$

Step 9.2.1.2

Divide $6$ by $1.50000007$ .

$f' (1.2247449) = - 1 \cdot 3.99999981 + 1$

Step 9.2.1.3

Multiply $- 1$ by $3.99999981$ .

$f' (1.2247449) = - 3.99999981 + 1$

Step 9.2.2

Add $- 3.99999981$ and $1$ .

$f' (1.2247449) = - 2.99999981$

Step 9.2.3

The final answer is $- 2.99999981$ .

$- 2.99999981$

Step 9.3

At $x = 1.2247449$ the derivative is $- 2.99999981$ . Since this is negative, the function is decreasing on $(0, \sqrt{6})$ .

Decreasing on $(0, \sqrt{6})$ since $f' (x) < 0$

Step 10

Substitute a value from the interval $(\sqrt{6}, \infty)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 10.1

Replace the variable $x$ with $3.4494898$ in the expression.

$f' (3.4494898) = - \frac{6}{{(3.4494898)}^{2}} + 1$

Step 10.2

Simplify the result.

Tap for more steps...

Step 10.2.1

Simplify each term.

Tap for more steps...

Step 10.2.1.1

Raise $3.4494898$ to the power of $2$ .

$f' (3.4494898) = - \frac{6}{11.89897988} + 1$

Step 10.2.1.2

Divide $6$ by $11.89897988$ .

$f' (3.4494898) = - 1 \cdot 0.5042449 + 1$

Step 10.2.1.3

Multiply $- 1$ by $0.5042449$ .

$f' (3.4494898) = - 0.5042449 + 1$

Step 10.2.2

Add $- 0.5042449$ and $1$ .

$f' (3.4494898) = 0.49575509$

Step 10.2.3

The final answer is $0.49575509$ .

$0.49575509$

Step 10.3

At $x = 3.4494898$ the derivative is $0.49575509$ . Since this is positive, the function is increasing on $(\sqrt{6}, \infty)$ .

Increasing on $(\sqrt{6}, \infty)$ since $f' (x) > 0$

Step 11

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

Increasing on: $(- \infty, - \sqrt{6}), (\sqrt{6}, \infty)$

Decreasing on: $(- \sqrt{6}, 0), (0, \sqrt{6})$

Step 12