Calculus Examples

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

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 $f (x) = x - 1$ and $g (x) = x^{2} + 1$ .

$\frac{(x^{2} + 1) \frac{d}{d x} [x - 1] - (x - 1) \frac{d}{d x} [x^{2} + 1]}{{(x^{2} + 1)}^{2}}$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

$\frac{(x^{2} + 1) (\frac{d}{d x} [x] + \frac{d}{d x} [- 1]) - (x - 1) \frac{d}{d x} [x^{2} + 1]}{{(x^{2} + 1)}^{2}}$

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 $n = 1$ .

$\frac{(x^{2} + 1) (1 + \frac{d}{d x} [- 1]) - (x - 1) \frac{d}{d x} [x^{2} + 1]}{{(x^{2} + 1)}^{2}}$

Step 1.1.2.3

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$\frac{(x^{2} + 1) (1 + 0) - (x - 1) \frac{d}{d x} [x^{2} + 1]}{{(x^{2} + 1)}^{2}}$

Step 1.1.2.4

Simplify the expression.

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Step 1.1.2.4.1

Add $1$ and $0$ .

$\frac{(x^{2} + 1) \cdot 1 - (x - 1) \frac{d}{d x} [x^{2} + 1]}{{(x^{2} + 1)}^{2}}$

Step 1.1.2.4.2

Multiply $x^{2} + 1$ by $1$ .

$\frac{x^{2} + 1 - (x - 1) \frac{d}{d x} [x^{2} + 1]}{{(x^{2} + 1)}^{2}}$

Step 1.1.2.5

By the Sum Rule, the derivative of $x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$ .

$\frac{x^{2} + 1 - (x - 1) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1])}{{(x^{2} + 1)}^{2}}$

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 $n = 2$ .

$\frac{x^{2} + 1 - (x - 1) (2 x + \frac{d}{d x} [1])}{{(x^{2} + 1)}^{2}}$

Step 1.1.2.7

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{x^{2} + 1 - (x - 1) (2 x + 0)}{{(x^{2} + 1)}^{2}}$

Step 1.1.2.8

Simplify the expression.

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Step 1.1.2.8.1

Add $2 x$ and $0$ .

$\frac{x^{2} + 1 - (x - 1) (2 x)}{{(x^{2} + 1)}^{2}}$

Step 1.1.2.8.2

Multiply $2$ by $- 1$ .

$\frac{x^{2} + 1 - 2 (x - 1) x}{{(x^{2} + 1)}^{2}}$

Step 1.1.3

Simplify.

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Step 1.1.3.1

Apply the distributive property.

$\frac{x^{2} + 1 + (- 2 x - 2 \cdot - 1) x}{{(x^{2} + 1)}^{2}}$

Step 1.1.3.2

Apply the distributive property.

$\frac{x^{2} + 1 - 2 x \cdot x - 2 \cdot - 1 x}{{(x^{2} + 1)}^{2}}$

Step 1.1.3.3

Simplify the numerator.

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Step 1.1.3.3.1

Simplify each term.

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Step 1.1.3.3.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 1.1.3.3.1.1.1

Move $x$ .

$\frac{x^{2} + 1 - 2 (x \cdot x) - 2 \cdot - 1 x}{{(x^{2} + 1)}^{2}}$

Step 1.1.3.3.1.1.2

Multiply $x$ by $x$ .

$\frac{x^{2} + 1 - 2 x^{2} - 2 \cdot - 1 x}{{(x^{2} + 1)}^{2}}$

Step 1.1.3.3.1.2

Multiply $- 2$ by $- 1$ .

$\frac{x^{2} + 1 - 2 x^{2} + 2 x}{{(x^{2} + 1)}^{2}}$

Step 1.1.3.3.2

Subtract $2 x^{2}$ from $x^{2}$ .

$\frac{- x^{2} + 1 + 2 x}{{(x^{2} + 1)}^{2}}$

Step 1.1.3.4

Reorder terms.

$\frac{- x^{2} + 2 x + 1}{{(x^{2} + 1)}^{2}}$

Step 1.1.3.5

Factor $- 1$ out of $- x^{2}$ .

$\frac{- (x^{2}) + 2 x + 1}{{(x^{2} + 1)}^{2}}$

Step 1.1.3.6

Factor $- 1$ out of $2 x$ .

$\frac{- (x^{2}) - (- 2 x) + 1}{{(x^{2} + 1)}^{2}}$

Step 1.1.3.7

Factor $- 1$ out of $- (x^{2}) - (- 2 x)$ .

$\frac{- (x^{2} - 2 x) + 1}{{(x^{2} + 1)}^{2}}$

Step 1.1.3.8

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

$\frac{- (x^{2} - 2 x) - 1 (- 1)}{{(x^{2} + 1)}^{2}}$

Step 1.1.3.9

Factor $- 1$ out of $- (x^{2} - 2 x) - 1 (- 1)$ .

$\frac{- (x^{2} - 2 x - 1)}{{(x^{2} + 1)}^{2}}$

Step 1.1.3.10

Rewrite $- (x^{2} - 2 x - 1)$ as $- 1 (x^{2} - 2 x - 1)$ .

$\frac{- 1 (x^{2} - 2 x - 1)}{{(x^{2} + 1)}^{2}}$

Step 1.1.3.11

Move the negative in front of the fraction.

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

Step 1.2

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

$- \frac{x^{2} - 2 x - 1}{{(x^{2} + 1)}^{2}}$

Step 2

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

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Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Set the numerator equal to zero.

$x^{2} - 2 x - 1 = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.3.2

Substitute the values $a = 1$ , $b = - 2$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (1 \cdot - 1)}}{2 \cdot 1}$

Step 2.3.3

Simplify.

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Step 2.3.3.1

Simplify the numerator.

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Step 2.3.3.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 1}}{2 \cdot 1}$

Step 2.3.3.1.2

Multiply $- 4 \cdot 1 \cdot - 1$ .

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Step 2.3.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot - 1}}{2 \cdot 1}$

Step 2.3.3.1.2.2

Multiply $- 4$ by $- 1$ .

$x = \frac{2 \pm \sqrt{4 + 4}}{2 \cdot 1}$

Step 2.3.3.1.3

Add $4$ and $4$ .

$x = \frac{2 \pm \sqrt{8}}{2 \cdot 1}$

Step 2.3.3.1.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 2.3.3.1.4.1

Factor $4$ out of $8$ .

$x = \frac{2 \pm \sqrt{4 (2)}}{2 \cdot 1}$

Step 2.3.3.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm \sqrt{2^{2} \cdot 2}}{2 \cdot 1}$

Step 2.3.3.1.5

Pull terms out from under the radical.

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

Step 2.3.3.2

Multiply $2$ by $1$ .

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

Step 2.3.3.3

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

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

Step 2.3.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.3.4.1

Simplify the numerator.

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Step 2.3.4.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 1}}{2 \cdot 1}$

Step 2.3.4.1.2

Multiply $- 4 \cdot 1 \cdot - 1$ .

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Step 2.3.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot - 1}}{2 \cdot 1}$

Step 2.3.4.1.2.2

Multiply $- 4$ by $- 1$ .

$x = \frac{2 \pm \sqrt{4 + 4}}{2 \cdot 1}$

Step 2.3.4.1.3

Add $4$ and $4$ .

$x = \frac{2 \pm \sqrt{8}}{2 \cdot 1}$

Step 2.3.4.1.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 2.3.4.1.4.1

Factor $4$ out of $8$ .

$x = \frac{2 \pm \sqrt{4 (2)}}{2 \cdot 1}$

Step 2.3.4.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm \sqrt{2^{2} \cdot 2}}{2 \cdot 1}$

Step 2.3.4.1.5

Pull terms out from under the radical.

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

Step 2.3.4.2

Multiply $2$ by $1$ .

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

Step 2.3.4.3

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

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

Step 2.3.4.4

Change the $\pm$ to $+$ .

$x = 1 + \sqrt{2}$

Step 2.3.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.3.5.1

Simplify the numerator.

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Step 2.3.5.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 1}}{2 \cdot 1}$

Step 2.3.5.1.2

Multiply $- 4 \cdot 1 \cdot - 1$ .

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Step 2.3.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot - 1}}{2 \cdot 1}$

Step 2.3.5.1.2.2

Multiply $- 4$ by $- 1$ .

$x = \frac{2 \pm \sqrt{4 + 4}}{2 \cdot 1}$

Step 2.3.5.1.3

Add $4$ and $4$ .

$x = \frac{2 \pm \sqrt{8}}{2 \cdot 1}$

Step 2.3.5.1.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 2.3.5.1.4.1

Factor $4$ out of $8$ .

$x = \frac{2 \pm \sqrt{4 (2)}}{2 \cdot 1}$

Step 2.3.5.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm \sqrt{2^{2} \cdot 2}}{2 \cdot 1}$

Step 2.3.5.1.5

Pull terms out from under the radical.

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

Step 2.3.5.2

Multiply $2$ by $1$ .

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

Step 2.3.5.3

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

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

Step 2.3.5.4

Change the $\pm$ to $-$ .

$x = 1 - \sqrt{2}$

Step 2.3.6

The final answer is the combination of both solutions.

$x = 1 + \sqrt{2}, 1 - \sqrt{2}$

Step 3

The values which make the derivative equal to $0$ are $1 + \sqrt{2}, 1 - \sqrt{2}$ .

$1 + \sqrt{2}, 1 - \sqrt{2}$

Step 4

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

$(- \infty, 1 - \sqrt{2}) \cup (1 - \sqrt{2}, 1 + \sqrt{2}) \cup (1 + \sqrt{2}, \infty)$

Step 5

Substitute a value from the interval $(- \infty, 1 - \sqrt{2})$ into the derivative to determine if the function is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 1.41421357$ in the expression.

$f' (- 1.41421357) = - \frac{{(- 1.41421357)}^{2} - 2 \cdot - 1.41421357 - 1}{{({(- 1.41421357)}^{2} + 1)}^{2}}$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify the numerator.

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Step 5.2.1.1

Raise $- 1.41421357$ to the power of $2$ .

$f' (- 1.41421357) = - \frac{2.00000002 - 2 \cdot - 1.41421357 - 1}{{({(- 1.41421357)}^{2} + 1)}^{2}}$

Step 5.2.1.2

Multiply $- 2$ by $- 1.41421357$ .

$f' (- 1.41421357) = - \frac{2.00000002 + 2.82842714 - 1}{{({(- 1.41421357)}^{2} + 1)}^{2}}$

Step 5.2.1.3

Add $2.00000002$ and $2.82842714$ .

$f' (- 1.41421357) = - \frac{4.82842716 - 1}{{({(- 1.41421357)}^{2} + 1)}^{2}}$

Step 5.2.1.4

Subtract $1$ from $4.82842716$ .

$f' (- 1.41421357) = - \frac{3.82842716}{{({(- 1.41421357)}^{2} + 1)}^{2}}$

Step 5.2.2

Simplify the denominator.

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Step 5.2.2.1

Raise $- 1.41421357$ to the power of $2$ .

$f' (- 1.41421357) = - \frac{3.82842716}{{(2.00000002 + 1)}^{2}}$

Step 5.2.2.2

Add $2.00000002$ and $1$ .

$f' (- 1.41421357) = - \frac{3.82842716}{{3.00000002}^{2}}$

Step 5.2.2.3

Raise $3.00000002$ to the power of $2$ .

$f' (- 1.41421357) = - \frac{3.82842716}{9.00000012}$

Step 5.2.3

Simplify the expression.

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Step 5.2.3.1

Divide $3.82842716$ by $9.00000012$ .

$f' (- 1.41421357) = - 1 \cdot 0.42538078$

Step 5.2.3.2

Multiply $- 1$ by $0.42538078$ .

$f' (- 1.41421357) = - 0.42538078$

Step 5.2.4

The final answer is $- 0.42538078$ .

$- 0.42538078$

Step 5.3

At $x = - 1.41421357$ the derivative is $- 0.42538078$ . Since this is negative, the function is decreasing on $(- \infty, 1 - \sqrt{2})$ .

Decreasing on $(- \infty, 1 - \sqrt{2})$ since $f' (x) < 0$

Step 6

Substitute a value from the interval $(- 0.41421357, 1 + \sqrt{2})$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $1.00000006$ in the expression.

$f' (1.00000006) = - \frac{{(1.00000006)}^{2} - 2 \cdot 1.00000006 - 1}{{({(1.00000006)}^{2} + 1)}^{2}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the numerator.

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Step 6.2.1.1

Raise $1.00000006$ to the power of $2$ .

$f' (1.00000006) = - \frac{1.00000013 - 2 \cdot 1.00000006 - 1}{{({1.00000006}^{2} + 1)}^{2}}$

Step 6.2.1.2

Multiply $- 2$ by $1.00000006$ .

$f' (1.00000006) = - \frac{1.00000013 - 2.00000013 - 1}{{({1.00000006}^{2} + 1)}^{2}}$

Step 6.2.1.3

Subtract $2.00000013$ from $1.00000013$ .

$f' (1.00000006) = - \frac{- 1 - 1}{{({1.00000006}^{2} + 1)}^{2}}$

Step 6.2.1.4

Subtract $1$ from $- 1$ .

$f' (1.00000006) = - \frac{- 2}{{({1.00000006}^{2} + 1)}^{2}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Raise $1.00000006$ to the power of $2$ .

$f' (1.00000006) = - \frac{- 2}{{(1.00000013 + 1)}^{2}}$

Step 6.2.2.2

Add $1.00000013$ and $1$ .

$f' (1.00000006) = - \frac{- 2}{{2.00000013}^{2}}$

Step 6.2.2.3

Raise $2.00000013$ to the power of $2$ .

$f' (1.00000006) = - \frac{- 2}{4.00000052}$

Step 6.2.3

Simplify the expression.

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Step 6.2.3.1

Divide $- 2$ by $4.00000052$ .

$f' (1.00000006) = 0.49999993$

Step 6.2.3.2

Multiply $- 1$ by $- 0.49999993$ .

$f' (1.00000006) = 0.49999993$

Step 6.2.4

The final answer is $0.49999993$ .

$0.49999993$

Step 6.3

At $x = 1.00000006$ the derivative is $0.49999993$ . Since this is positive, the function is increasing on $(- 0.41421357, 1 + \sqrt{2})$ .

Increasing on $(1 - \sqrt{2}, 1 + \sqrt{2})$ since $f' (x) > 0$

Step 7

Substitute a value from the interval $(1 + \sqrt{2}, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $3.4142137$ in the expression.

$f' (3.4142137) = - \frac{{(3.4142137)}^{2} - 2 \cdot 3.4142137 - 1}{{({(3.4142137)}^{2} + 1)}^{2}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify the numerator.

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Step 7.2.1.1

Raise $3.4142137$ to the power of $2$ .

$f' (3.4142137) = - \frac{11.65685518 - 2 \cdot 3.4142137 - 1}{{({3.4142137}^{2} + 1)}^{2}}$

Step 7.2.1.2

Multiply $- 2$ by $3.4142137$ .

$f' (3.4142137) = - \frac{11.65685518 - 6.8284274 - 1}{{({3.4142137}^{2} + 1)}^{2}}$

Step 7.2.1.3

Subtract $6.8284274$ from $11.65685518$ .

$f' (3.4142137) = - \frac{4.82842778 - 1}{{({3.4142137}^{2} + 1)}^{2}}$

Step 7.2.1.4

Subtract $1$ from $4.82842778$ .

$f' (3.4142137) = - \frac{3.82842778}{{({3.4142137}^{2} + 1)}^{2}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

Raise $3.4142137$ to the power of $2$ .

$f' (3.4142137) = - \frac{3.82842778}{{(11.65685518 + 1)}^{2}}$

Step 7.2.2.2

Add $11.65685518$ and $1$ .

$f' (3.4142137) = - \frac{3.82842778}{{12.65685518}^{2}}$

Step 7.2.2.3

Raise $12.65685518$ to the power of $2$ .

$f' (3.4142137) = - \frac{3.82842778}{160.19598328}$

Step 7.2.3

Simplify the expression.

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Step 7.2.3.1

Divide $3.82842778$ by $160.19598328$ .

$f' (3.4142137) = - 1 \cdot 0.0238984$

Step 7.2.3.2

Multiply $- 1$ by $0.0238984$ .

$f' (3.4142137) = - 0.0238984$

Step 7.2.4

The final answer is $- 0.0238984$ .

$- 0.0238984$

Step 7.3

At $x = 3.4142137$ the derivative is $- 0.0238984$ . Since this is negative, the function is decreasing on $(1 + \sqrt{2}, \infty)$ .

Decreasing on $(1 + \sqrt{2}, \infty)$ since $f' (x) < 0$

Step 8

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

Increasing on: $(1 - \sqrt{2}, 1 + \sqrt{2})$

Decreasing on: $(- \infty, 1 - \sqrt{2}), (1 + \sqrt{2}, \infty)$

Step 9