Calculus Examples

$x e^{x}$

Step 1

Write $x e^{x}$ as a function.

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

Step 2

Find the second derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

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 $g (x) = e^{x}$ .

$x \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x]$

Step 2.1.2

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

$x e^{x} + e^{x} \frac{d}{d x} [x]$

Step 2.1.3

Differentiate using the Power Rule.

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Step 2.1.3.1

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

$x e^{x} + e^{x} \cdot 1$

Step 2.1.3.2

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

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

Step 2.2

Find the second derivative.

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Step 2.2.1

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

$\frac{d}{d x} [x e^{x}] + \frac{d}{d x} [e^{x}]$

Step 2.2.2

Evaluate $\frac{d}{d x} [x e^{x}]$ .

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Step 2.2.2.1

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 $g (x) = e^{x}$ .

$x \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x] + \frac{d}{d x} [e^{x}]$

Step 2.2.2.2

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

$x e^{x} + e^{x} \frac{d}{d x} [x] + \frac{d}{d x} [e^{x}]$

Step 2.2.2.3

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

$x e^{x} + e^{x} \cdot 1 + \frac{d}{d x} [e^{x}]$

Step 2.2.2.4

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

$x e^{x} + e^{x} + \frac{d}{d x} [e^{x}]$

Step 2.2.3

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

$x e^{x} + e^{x} + e^{x}$

Step 2.2.4

Simplify.

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Step 2.2.4.1

Add $e^{x}$ and $e^{x}$ .

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

Step 2.2.4.2

Reorder terms.

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

Step 2.2.4.3

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

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

Step 2.3

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

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

Step 3

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

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Step 3.1

Set the second derivative equal to $0$ .

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

$e^{x} x + e^{x} \cdot 2 = 0$

Step 3.2.3

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

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

Step 3.3

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

$e^{x} = 0$

$x + 2 = 0$

Step 3.4

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

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Step 3.4.1

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

$e^{x} = 0$

Step 3.4.2

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

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Step 3.4.2.1

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

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

Step 3.4.2.2

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

Undefined

Step 3.4.2.3

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

No solution

Step 3.5

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 3.5.1

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

$x + 2 = 0$

Step 3.5.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 3.6

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

$x = - 2$

Step 4

Find the points where the second derivative is $0$ .

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Step 4.1

Substitute $- 2$ in $f (x) = x e^{x}$ to find the value of $y$ .

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Step 4.1.1

Replace the variable $x$ with $- 2$ in the expression.

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

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

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

$f (- 2) = - 2 \cdot \frac{1}{e^{2}}$

Step 4.1.2.2

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

$f (- 2) = \frac{- 2}{e^{2}}$

Step 4.1.2.3

Move the negative in front of the fraction.

$f (- 2) = - \frac{2}{e^{2}}$

Step 4.1.2.4

The final answer is $- \frac{2}{e^{2}}$ .

$- \frac{2}{e^{2}}$

Step 4.2

The point found by substituting $- 2$ in $f (x) = x e^{x}$ is $(- 2, - \frac{2}{e^{2}})$ . This point can be an inflection point.

$(- 2, - \frac{2}{e^{2}})$

Step 5

Split $(- \infty, \infty)$ into intervals around the points that could potentially be inflection points.

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

Step 6

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

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Step 6.1

Replace the variable $x$ with $- 2.1$ in the expression.

$f'' (- 2.1) = (- 2.1) \cdot e^{- 2.1} + 2 e^{- 2.1}$

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

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

$f'' (- 2.1) = - 2.1 \cdot \frac{1}{e^{2.1}} + 2 e^{- 2.1}$

Step 6.2.1.2

Combine $- 2.1$ and $\frac{1}{e^{2.1}}$ .

$f'' (- 2.1) = \frac{- 2.1}{e^{2.1}} + 2 e^{- 2.1}$

Step 6.2.1.3

Move the negative in front of the fraction.

$f'' (- 2.1) = - \frac{2.1}{e^{2.1}} + 2 e^{- 2.1}$

Step 6.2.1.4

Replace $e$ with an approximation.

$f'' (- 2.1) = - \frac{2.1}{{2.71828182}^{2.1}} + 2 e^{- 2.1}$

Step 6.2.1.5

Raise $2.71828182$ to the power of $2.1$ .

$f'' (- 2.1) = - \frac{2.1}{8.16616991} + 2 e^{- 2.1}$

Step 6.2.1.6

Divide $2.1$ by $8.16616991$ .

$f'' (- 2.1) = - 1 \cdot 0.25715849 + 2 e^{- 2.1}$

Step 6.2.1.7

Multiply $- 1$ by $0.25715849$ .

$f'' (- 2.1) = - 0.25715849 + 2 e^{- 2.1}$

Step 6.2.1.8

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

$f'' (- 2.1) = - 0.25715849 + 2 (\frac{1}{e^{2.1}})$

Step 6.2.1.9

Combine $2$ and $\frac{1}{e^{2.1}}$ .

$f'' (- 2.1) = - 0.25715849 + \frac{2}{e^{2.1}}$

Step 6.2.2

Add $- 0.25715849$ and $\frac{2}{e^{2.1}}$ .

$f'' (- 2.1) = - 0.01224564$

Step 6.2.3

The final answer is $- 0.01224564$ .

$- 0.01224564$

Step 6.3

At $- 2.1$ , the second derivative is $- 0.01224564$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, - 2)$

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

Step 7

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

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Step 7.1

Replace the variable $x$ with $- 1.9$ in the expression.

$f'' (- 1.9) = (- 1.9) \cdot e^{- 1.9} + 2 e^{- 1.9}$

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

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

$f'' (- 1.9) = - 1.9 \cdot \frac{1}{e^{1.9}} + 2 e^{- 1.9}$

Step 7.2.1.2

Combine $- 1.9$ and $\frac{1}{e^{1.9}}$ .

$f'' (- 1.9) = \frac{- 1.9}{e^{1.9}} + 2 e^{- 1.9}$

Step 7.2.1.3

Move the negative in front of the fraction.

$f'' (- 1.9) = - \frac{1.9}{e^{1.9}} + 2 e^{- 1.9}$

Step 7.2.1.4

Replace $e$ with an approximation.

$f'' (- 1.9) = - \frac{1.9}{{2.71828182}^{1.9}} + 2 e^{- 1.9}$

Step 7.2.1.5

Raise $2.71828182$ to the power of $1.9$ .

$f'' (- 1.9) = - \frac{1.9}{6.68589444} + 2 e^{- 1.9}$

Step 7.2.1.6

Divide $1.9$ by $6.68589444$ .

$f'' (- 1.9) = - 1 \cdot 0.28418037 + 2 e^{- 1.9}$

Step 7.2.1.7

Multiply $- 1$ by $0.28418037$ .

$f'' (- 1.9) = - 0.28418037 + 2 e^{- 1.9}$

Step 7.2.1.8

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

$f'' (- 1.9) = - 0.28418037 + 2 (\frac{1}{e^{1.9}})$

Step 7.2.1.9

Combine $2$ and $\frac{1}{e^{1.9}}$ .

$f'' (- 1.9) = - 0.28418037 + \frac{2}{e^{1.9}}$

Step 7.2.2

Add $- 0.28418037$ and $\frac{2}{e^{1.9}}$ .

$f'' (- 1.9) = 0.01495686$

Step 7.2.3

The final answer is $0.01495686$ .

$0.01495686$

Step 7.3

At $- 1.9$ , the second derivative is $0.01495686$ . Since this is positive, the second derivative is increasing on the interval $(- 2, \infty)$ .

Increasing on $(- 2, \infty)$ since $f'' (x) > 0$

Step 8

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection point in this case is $(- 2, - \frac{2}{e^{2}})$ .

$(- 2, - \frac{2}{e^{2}})$

Step 9