Calculus Examples

$f (x) = x e^{- x^{2}}$

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 Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = e^{- x^{2}}$ .

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

Step 1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = - x^{2}$ .

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Step 1.1.2.1

To apply the Chain Rule, set $u$ as $- x^{2}$ .

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

Step 1.1.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

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

Step 1.1.2.3

Replace all occurrences of $u$ with $- x^{2}$ .

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

Step 1.1.3

Differentiate.

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Step 1.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

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

Step 1.1.3.2

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

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

Step 1.1.3.3

Multiply $2$ by $- 1$ .

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

Step 1.1.4

Raise $x$ to the power of $1$ .

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

Step 1.1.5

Raise $x$ to the power of $1$ .

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

Step 1.1.6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.1.7

Simplify the expression.

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Step 1.1.7.1

Add $1$ and $1$ .

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

Step 1.1.7.2

Move $- 2$ to the left of $e^{- x^{2}}$ .

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

Step 1.1.8

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

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

Step 1.1.9

Multiply $e^{- x^{2}}$ by $1$ .

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

Step 1.1.10

Simplify.

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Step 1.1.10.1

Reorder terms.

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

Step 1.1.10.2

Reorder factors in $- 2 e^{- x^{2}} x^{2} + e^{- x^{2}}$ .

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

Step 1.2

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

$- 2 x^{2} e^{- x^{2}} + e^{- x^{2}}$

Step 2

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

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Step 2.1

Set the first derivative equal to $0$ .

$- 2 x^{2} e^{- x^{2}} + e^{- x^{2}} = 0$

Step 2.2

Factor $e^{- x^{2}}$ out of $- 2 x^{2} e^{- x^{2}} + e^{- x^{2}}$ .

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Step 2.2.1

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

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

Step 2.2.2

Multiply by $1$ .

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

Step 2.2.3

Factor $e^{- x^{2}}$ out of $e^{- x^{2}} (- 2 x^{2}) + e^{- x^{2}} \cdot 1$ .

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

Step 2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

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

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

Step 2.4

Set $e^{- x^{2}}$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $e^{- x^{2}}$ equal to $0$ .

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

Step 2.4.2

Solve $e^{- x^{2}} = 0$ for $x$ .

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Step 2.4.2.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{- x^{2}}) = \ln (0)$

Step 2.4.2.2

The equation cannot be solved because $\ln (0)$ is undefined.

Undefined

Step 2.4.2.3

There is no solution for $e^{- x^{2}} = 0$

No solution

Step 2.5

Set $- 2 x^{2} + 1$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $- 2 x^{2} + 1$ equal to $0$ .

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

Step 2.5.2

Solve $- 2 x^{2} + 1 = 0$ for $x$ .

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Step 2.5.2.1

Subtract $1$ from both sides of the equation.

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

Step 2.5.2.2

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

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Step 2.5.2.2.1

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

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

Step 2.5.2.2.2

Simplify the left side.

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Step 2.5.2.2.2.1

Cancel the common factor of $- 2$ .

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Step 2.5.2.2.2.1.1

Cancel the common factor.

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

Step 2.5.2.2.2.1.2

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

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

Step 2.5.2.2.3

Simplify the right side.

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Step 2.5.2.2.3.1

Dividing two negative values results in a positive value.

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

Step 2.5.2.3

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

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

Step 2.5.2.4

Simplify $\pm \sqrt{\frac{1}{2}}$ .

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Step 2.5.2.4.1

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

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

Step 2.5.2.4.2

Any root of $1$ is $1$ .

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

Step 2.5.2.4.3

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 2.5.2.4.4

Combine and simplify the denominator.

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Step 2.5.2.4.4.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 2.5.2.4.4.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 2.5.2.4.4.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 2.5.2.4.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.5.2.4.4.5

Add $1$ and $1$ .

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

Step 2.5.2.4.4.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 2.5.2.4.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

Step 2.5.2.4.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.5.2.4.4.6.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 2.5.2.4.4.6.4

Cancel the common factor of $2$ .

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Step 2.5.2.4.4.6.4.1

Cancel the common factor.

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

Step 2.5.2.4.4.6.4.2

Rewrite the expression.

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

Step 2.5.2.4.4.6.5

Evaluate the exponent.

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

Step 2.5.2.5

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

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Step 2.5.2.5.1

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

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

Step 2.5.2.5.2

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

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

Step 2.5.2.5.3

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

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

Step 2.6

The final solution is all the values that make $e^{- x^{2}} (- 2 x^{2} + 1) = 0$ true.

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

Step 3

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

$\frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2}$

Step 4

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

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

Step 5

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

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Step 5.1

Replace the variable $x$ with $- 1.70710677$ in the expression.

$f' (- 1.70710677) = - 2 {(- 1.70710677)}^{2} e^{- {(- 1.70710677)}^{2}} + e^{- {(- 1.70710677)}^{2}}$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Raise $- 1.70710677$ to the power of $2$ .

$f' (- 1.70710677) = - 2 \cdot (2.91421352 e^{- {(- 1.70710677)}^{2}}) + e^{- {(- 1.70710677)}^{2}}$

Step 5.2.1.2

Multiply $- 2$ by $2.91421352$ .

$f' (- 1.70710677) = - 5.82842704 e^{- {(- 1.70710677)}^{2}} + e^{- {(- 1.70710677)}^{2}}$

Step 5.2.1.3

Raise $- 1.70710677$ to the power of $2$ .

$f' (- 1.70710677) = - 5.82842704 e^{- 1 \cdot 2.91421352} + e^{- {(- 1.70710677)}^{2}}$

Step 5.2.1.4

Multiply $- 1$ by $2.91421352$ .

$f' (- 1.70710677) = - 5.82842704 e^{- 2.91421352} + e^{- {(- 1.70710677)}^{2}}$

Step 5.2.1.5

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

$f' (- 1.70710677) = - 5.82842704 \frac{1}{e^{2.91421352}} + e^{- {(- 1.70710677)}^{2}}$

Step 5.2.1.6

Combine $- 5.82842704$ and $\frac{1}{e^{2.91421352}}$ .

$f' (- 1.70710677) = \frac{- 5.82842704}{e^{2.91421352}} + e^{- {(- 1.70710677)}^{2}}$

Step 5.2.1.7

Move the negative in front of the fraction.

$f' (- 1.70710677) = - \frac{5.82842704}{e^{2.91421352}} + e^{- {(- 1.70710677)}^{2}}$

Step 5.2.1.8

Replace $e$ with an approximation.

$f' (- 1.70710677) = - \frac{5.82842704}{{2.71828182}^{2.91421352}} + e^{- {(- 1.70710677)}^{2}}$

Step 5.2.1.9

Raise $2.71828182$ to the power of $2.91421352$ .

$f' (- 1.70710677) = - \frac{5.82842704}{18.43430856} + e^{- {(- 1.70710677)}^{2}}$

Step 5.2.1.10

Divide $5.82842704$ by $18.43430856$ .

$f' (- 1.70710677) = - 1 \cdot 0.3161728 + e^{- {(- 1.70710677)}^{2}}$

Step 5.2.1.11

Multiply $- 1$ by $0.3161728$ .

$f' (- 1.70710677) = - 0.3161728 + e^{- {(- 1.70710677)}^{2}}$

Step 5.2.1.12

Raise $- 1.70710677$ to the power of $2$ .

$f' (- 1.70710677) = - 0.3161728 + e^{- 1 \cdot 2.91421352}$

Step 5.2.1.13

Multiply $- 1$ by $2.91421352$ .

$f' (- 1.70710677) = - 0.3161728 + e^{- 2.91421352}$

Step 5.2.1.14

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

$f' (- 1.70710677) = - 0.3161728 + \frac{1}{e^{2.91421352}}$

Step 5.2.2

Add $- 0.3161728$ and $\frac{1}{e^{2.91421352}}$ .

$f' (- 1.70710677) = - 0.26192612$

Step 5.2.3

The final answer is $- 0.26192612$ .

$- 0.26192612$

Step 5.3

At $x = - 1.70710677$ the derivative is $- 0.26192612$ . Since this is negative, the function is decreasing on $(- \infty, - \frac{\sqrt{2}}{2})$ .

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

Step 6

Substitute a value from the interval $(- 0.70710677, \frac{\sqrt{2}}{2})$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $0$ in the expression.

$f' (0) = - 2 {(0)}^{2} e^{- {(0)}^{2}} + e^{- {(0)}^{2}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Raising $0$ to any positive power yields $0$ .

$f' (0) = - 2 \cdot (0 e^{- {(0)}^{2}}) + e^{- {(0)}^{2}}$

Step 6.2.1.2

Multiply $- 2$ by $0$ .

$f' (0) = 0 e^{- {(0)}^{2}} + e^{- {(0)}^{2}}$

Step 6.2.1.3

Raising $0$ to any positive power yields $0$ .

$f' (0) = 0 e^{- 0} + e^{- {(0)}^{2}}$

Step 6.2.1.4

Multiply $- 1$ by $0$ .

$f' (0) = 0 e^{0} + e^{- {(0)}^{2}}$

Step 6.2.1.5

Anything raised to $0$ is $1$ .

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

Step 6.2.1.6

Multiply $0$ by $1$ .

$f' (0) = 0 + e^{- {(0)}^{2}}$

Step 6.2.1.7

Raising $0$ to any positive power yields $0$ .

$f' (0) = 0 + e^{- 0}$

Step 6.2.1.8

Multiply $- 1$ by $0$ .

$f' (0) = 0 + e^{0}$

Step 6.2.1.9

Anything raised to $0$ is $1$ .

$f' (0) = 0 + 1$

Step 6.2.2

Add $0$ and $1$ .

$f' (0) = 1$

Step 6.2.3

The final answer is $1$ .

$1$

Step 6.3

At $x = 0$ the derivative is $1$ . Since this is positive, the function is increasing on $(- 0.70710677, \frac{\sqrt{2}}{2})$ .

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

Step 7

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

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Step 7.1

Replace the variable $x$ with $1.70710677$ in the expression.

$f' (1.70710677) = - 2 {(1.70710677)}^{2} e^{- {(1.70710677)}^{2}} + e^{- {(1.70710677)}^{2}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Raise $1.70710677$ to the power of $2$ .

$f' (1.70710677) = - 2 \cdot (2.91421352 e^{- {(1.70710677)}^{2}}) + e^{- {(1.70710677)}^{2}}$

Step 7.2.1.2

Multiply $- 2$ by $2.91421352$ .

$f' (1.70710677) = - 5.82842704 e^{- {(1.70710677)}^{2}} + e^{- {(1.70710677)}^{2}}$

Step 7.2.1.3

Raise $1.70710677$ to the power of $2$ .

$f' (1.70710677) = - 5.82842704 e^{- 1 \cdot 2.91421352} + e^{- {(1.70710677)}^{2}}$

Step 7.2.1.4

Multiply $- 1$ by $2.91421352$ .

$f' (1.70710677) = - 5.82842704 e^{- 2.91421352} + e^{- {(1.70710677)}^{2}}$

Step 7.2.1.5

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

$f' (1.70710677) = - 5.82842704 \frac{1}{e^{2.91421352}} + e^{- {(1.70710677)}^{2}}$

Step 7.2.1.6

Combine $- 5.82842704$ and $\frac{1}{e^{2.91421352}}$ .

$f' (1.70710677) = \frac{- 5.82842704}{e^{2.91421352}} + e^{- {(1.70710677)}^{2}}$

Step 7.2.1.7

Move the negative in front of the fraction.

$f' (1.70710677) = - \frac{5.82842704}{e^{2.91421352}} + e^{- {(1.70710677)}^{2}}$

Step 7.2.1.8

Replace $e$ with an approximation.

$f' (1.70710677) = - \frac{5.82842704}{{2.71828182}^{2.91421352}} + e^{- {(1.70710677)}^{2}}$

Step 7.2.1.9

Raise $2.71828182$ to the power of $2.91421352$ .

$f' (1.70710677) = - \frac{5.82842704}{18.43430856} + e^{- {(1.70710677)}^{2}}$

Step 7.2.1.10

Divide $5.82842704$ by $18.43430856$ .

$f' (1.70710677) = - 1 \cdot 0.3161728 + e^{- {(1.70710677)}^{2}}$

Step 7.2.1.11

Multiply $- 1$ by $0.3161728$ .

$f' (1.70710677) = - 0.3161728 + e^{- {(1.70710677)}^{2}}$

Step 7.2.1.12

Raise $1.70710677$ to the power of $2$ .

$f' (1.70710677) = - 0.3161728 + e^{- 1 \cdot 2.91421352}$

Step 7.2.1.13

Multiply $- 1$ by $2.91421352$ .

$f' (1.70710677) = - 0.3161728 + e^{- 2.91421352}$

Step 7.2.1.14

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

$f' (1.70710677) = - 0.3161728 + \frac{1}{e^{2.91421352}}$

Step 7.2.2

Add $- 0.3161728$ and $\frac{1}{e^{2.91421352}}$ .

$f' (1.70710677) = - 0.26192612$

Step 7.2.3

The final answer is $- 0.26192612$ .

$- 0.26192612$

Step 7.3

At $x = 1.70710677$ the derivative is $- 0.26192612$ . Since this is negative, the function is decreasing on $(\frac{\sqrt{2}}{2}, \infty)$ .

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

Step 8

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

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

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

Step 9