Calculus Examples

$x^{3} e^{x}$

Step 1

Find the first derivative of the function.

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Step 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 $f (x) = x^{3}$ and $g (x) = e^{x}$ .

$x^{3} \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{3}]$

Step 1.2

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

$x^{3} e^{x} + e^{x} \frac{d}{d x} [x^{3}]$

Step 1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$x^{3} e^{x} + e^{x} (3 x^{2})$

Step 1.4

Simplify.

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Step 1.4.1

Reorder terms.

$e^{x} x^{3} + 3 e^{x} x^{2}$

Step 1.4.2

Reorder factors in $e^{x} x^{3} + 3 e^{x} x^{2}$ .

$x^{3} e^{x} + 3 x^{2} e^{x}$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $x^{3} e^{x} + 3 x^{2} e^{x}$ with respect to $x$ is $\frac{d}{d x} [x^{3} e^{x}] + \frac{d}{d x} [3 x^{2} e^{x}]$ .

$f'' (x) = \frac{d}{d x} (x^{3} e^{x}) + \frac{d}{d x} (3 x^{2} e^{x})$

Step 2.2

Evaluate $\frac{d}{d x} [x^{3} e^{x}]$ .

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Step 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 $f (x) = x^{3}$ and $g (x) = e^{x}$ .

$f'' (x) = x^{3} \frac{d}{d x} (e^{x}) + e^{x} \frac{d}{d x} (x^{3}) + \frac{d}{d x} (3 x^{2} e^{x})$

Step 2.2.2

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

$f'' (x) = x^{3} e^{x} + e^{x} \frac{d}{d x} (x^{3}) + \frac{d}{d x} (3 x^{2} e^{x})$

Step 2.2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$f'' (x) = x^{3} e^{x} + e^{x} (3 x^{2}) + \frac{d}{d x} (3 x^{2} e^{x})$

Step 2.3

Evaluate $\frac{d}{d x} [3 x^{2} e^{x}]$ .

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Step 2.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2} e^{x}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2} e^{x}]$ .

$f'' (x) = x^{3} e^{x} + e^{x} (3 x^{2}) + 3 \frac{d}{d x} (x^{2} e^{x})$

Step 2.3.2

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 $f (x) = x^{2}$ and $g (x) = e^{x}$ .

$f'' (x) = x^{3} e^{x} + e^{x} (3 x^{2}) + 3 (x^{2} \frac{d}{d x} (e^{x}) + e^{x} \frac{d}{d x} (x^{2}))$

Step 2.3.3

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

$f'' (x) = x^{3} e^{x} + e^{x} (3 x^{2}) + 3 (x^{2} e^{x} + e^{x} \frac{d}{d x} (x^{2}))$

Step 2.3.4

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

$f'' (x) = x^{3} e^{x} + e^{x} (3 x^{2}) + 3 (x^{2} e^{x} + e^{x} (2 x))$

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$f'' (x) = x^{3} e^{x} + e^{x} (3 x^{2}) + 3 (x^{2} e^{x}) + 3 (e^{x} (2 x))$

Step 2.4.2

Combine terms.

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Step 2.4.2.1

Multiply $2$ by $3$ .

$f'' (x) = x^{3} e^{x} + e^{x} (3 x^{2}) + 3 x^{2} e^{x} + 6 (e^{x} (x))$

Step 2.4.2.2

Add $e^{x} (3 x^{2})$ and $3 x^{2} e^{x}$ .

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Step 2.4.2.2.1

Move $e^{x}$ .

$f'' (x) = x^{3} e^{x} + 3 x^{2} e^{x} + 3 x^{2} e^{x} + 6 e^{x} x$

Step 2.4.2.2.2

Add $3 x^{2} e^{x}$ and $3 x^{2} e^{x}$ .

$f'' (x) = x^{3} e^{x} + 6 x^{2} e^{x} + 6 e^{x} x$

Step 2.4.3

Reorder terms.

$f'' (x) = e^{x} x^{3} + 6 e^{x} x^{2} + 6 e^{x} x$

Step 2.4.4

Reorder factors in $e^{x} x^{3} + 6 e^{x} x^{2} + 6 e^{x} x$ .

$f'' (x) = x^{3} e^{x} + 6 x^{2} e^{x} + 6 x e^{x}$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$x^{3} e^{x} + 3 x^{2} e^{x} = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

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 $f (x) = x^{3}$ and $g (x) = e^{x}$ .

$x^{3} \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{3}]$

Step 4.1.2

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

$x^{3} e^{x} + e^{x} \frac{d}{d x} [x^{3}]$

Step 4.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$x^{3} e^{x} + e^{x} (3 x^{2})$

Step 4.1.4

Simplify.

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Step 4.1.4.1

Reorder terms.

$e^{x} x^{3} + 3 e^{x} x^{2}$

Step 4.1.4.2

Reorder factors in $e^{x} x^{3} + 3 e^{x} x^{2}$ .

$f' (x) = x^{3} e^{x} + 3 x^{2} e^{x}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $x^{3} e^{x} + 3 x^{2} e^{x}$ .

$x^{3} e^{x} + 3 x^{2} e^{x}$

Step 5

Set the first derivative equal to $0$ then solve the equation $x^{3} e^{x} + 3 x^{2} e^{x} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$x^{3} e^{x} + 3 x^{2} e^{x} = 0$

Step 5.2

Factor $x^{2} e^{x}$ out of $x^{3} e^{x} + 3 x^{2} e^{x}$ .

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Step 5.2.1

Factor $x^{2} e^{x}$ out of $x^{3} e^{x}$ .

$x^{2} e^{x} (x) + 3 x^{2} e^{x} = 0$

Step 5.2.2

Factor $x^{2} e^{x}$ out of $3 x^{2} e^{x}$ .

$x^{2} e^{x} (x) + x^{2} e^{x} (3) = 0$

Step 5.2.3

Factor $x^{2} e^{x}$ out of $x^{2} e^{x} (x) + x^{2} e^{x} (3)$ .

$x^{2} e^{x} (x + 3) = 0$

Step 5.3

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

$x^{2} = 0$

$e^{x} = 0$

$x + 3 = 0$

Step 5.4

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

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Step 5.4.1

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

$x^{2} = 0$

Step 5.4.2

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

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Step 5.4.2.1

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

$x = \pm \sqrt{0}$

Step 5.4.2.2

Simplify $\pm \sqrt{0}$ .

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Step 5.4.2.2.1

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

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

Step 5.4.2.2.2

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

$x = \pm 0$

Step 5.4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 5.5

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

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Step 5.5.1

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

$e^{x} = 0$

Step 5.5.2

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

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Step 5.5.2.1

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

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

Step 5.5.2.2

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

Undefined

Step 5.5.2.3

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

No solution

Step 5.6

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 5.6.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 5.6.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 5.7

The final solution is all the values that make $x^{2} e^{x} (x + 3) = 0$ true.

$x = 0, - 3$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 7

Critical points to evaluate.

$x = 0, - 3$

Step 8

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

${(0)}^{3} e^{0} + 6 {(0)}^{2} e^{0} + 6 (0) e^{0}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify each term.

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Step 9.1.1

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

$0 e^{0} + 6 {(0)}^{2} e^{0} + 6 (0) e^{0}$

Step 9.1.2

Anything raised to $0$ is $1$ .

$0 \cdot 1 + 6 {(0)}^{2} e^{0} + 6 (0) e^{0}$

Step 9.1.3

Multiply $0$ by $1$ .

$0 + 6 {(0)}^{2} e^{0} + 6 (0) e^{0}$

Step 9.1.4

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

$0 + 6 \cdot 0 e^{0} + 6 (0) e^{0}$

Step 9.1.5

Multiply $6$ by $0$ .

$0 + 0 e^{0} + 6 (0) e^{0}$

Step 9.1.6

Anything raised to $0$ is $1$ .

$0 + 0 \cdot 1 + 6 (0) e^{0}$

Step 9.1.7

Multiply $0$ by $1$ .

$0 + 0 + 6 (0) e^{0}$

Step 9.1.8

Multiply $6$ by $0$ .

$0 + 0 + 0 e^{0}$

Step 9.1.9

Anything raised to $0$ is $1$ .

$0 + 0 + 0 \cdot 1$

Step 9.1.10

Multiply $0$ by $1$ .

$0 + 0 + 0$

Step 9.2

Simplify by adding numbers.

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Step 9.2.1

Add $0$ and $0$ .

$0 + 0$

Step 9.2.2

Add $0$ and $0$ .

$0$

Step 10

Since there is at least one point with $0$ or undefined second derivative, apply the first derivative test.

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Step 10.1

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

$(- \infty, - 3) \cup (- 3, 0) \cup (0, \infty)$

Step 10.2

Substitute any number, such as $- 6$ , from the interval $(- \infty, - 3)$ in the first derivative $x^{3} e^{x} + 3 x^{2} e^{x}$ to check if the result is negative or positive.

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Step 10.2.1

Replace the variable $x$ with $- 6$ in the expression.

$f' (- 6) = {(- 6)}^{3} e^{- 6} + 3 {(- 6)}^{2} e^{- 6}$

Step 10.2.2

Simplify the result.

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Step 10.2.2.1

Simplify each term.

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Step 10.2.2.1.1

Raise $- 6$ to the power of $3$ .

$f' (- 6) = - 216 e^{- 6} + 3 {(- 6)}^{2} e^{- 6}$

Step 10.2.2.1.2

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

$f' (- 6) = - 216 \frac{1}{e^{6}} + 3 {(- 6)}^{2} e^{- 6}$

Step 10.2.2.1.3

Combine $- 216$ and $\frac{1}{e^{6}}$ .

$f' (- 6) = \frac{- 216}{e^{6}} + 3 {(- 6)}^{2} e^{- 6}$

Step 10.2.2.1.4

Move the negative in front of the fraction.

$f' (- 6) = - \frac{216}{e^{6}} + 3 {(- 6)}^{2} e^{- 6}$

Step 10.2.2.1.5

Raise $- 6$ to the power of $2$ .

$f' (- 6) = - \frac{216}{e^{6}} + 3 \cdot (36 e^{- 6})$

Step 10.2.2.1.6

Multiply $3$ by $36$ .

$f' (- 6) = - \frac{216}{e^{6}} + 108 e^{- 6}$

Step 10.2.2.1.7

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

$f' (- 6) = - \frac{216}{e^{6}} + 108 (\frac{1}{e^{6}})$

Step 10.2.2.1.8

Combine $108$ and $\frac{1}{e^{6}}$ .

$f' (- 6) = - \frac{216}{e^{6}} + \frac{108}{e^{6}}$

Step 10.2.2.2

Combine fractions.

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Step 10.2.2.2.1

Combine the numerators over the common denominator.

$f' (- 6) = \frac{- 216 + 108}{e^{6}}$

Step 10.2.2.2.2

Simplify the expression.

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Step 10.2.2.2.2.1

Add $- 216$ and $108$ .

$f' (- 6) = \frac{- 108}{e^{6}}$

Step 10.2.2.2.2.2

Move the negative in front of the fraction.

$f' (- 6) = - \frac{108}{e^{6}}$

Step 10.2.2.3

The final answer is $- \frac{108}{e^{6}}$ .

$- \frac{108}{e^{6}}$

Step 10.3

Substitute any number, such as $- 2$ , from the interval $(- 3, 0)$ in the first derivative $x^{3} e^{x} + 3 x^{2} e^{x}$ to check if the result is negative or positive.

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Step 10.3.1

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

$f' (- 2) = {(- 2)}^{3} e^{- 2} + 3 {(- 2)}^{2} e^{- 2}$

Step 10.3.2

Simplify the result.

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Step 10.3.2.1

Simplify each term.

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Step 10.3.2.1.1

Raise $- 2$ to the power of $3$ .

$f' (- 2) = - 8 e^{- 2} + 3 {(- 2)}^{2} e^{- 2}$

Step 10.3.2.1.2

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

$f' (- 2) = - 8 \frac{1}{e^{2}} + 3 {(- 2)}^{2} e^{- 2}$

Step 10.3.2.1.3

Combine $- 8$ and $\frac{1}{e^{2}}$ .

$f' (- 2) = \frac{- 8}{e^{2}} + 3 {(- 2)}^{2} e^{- 2}$

Step 10.3.2.1.4

Move the negative in front of the fraction.

$f' (- 2) = - \frac{8}{e^{2}} + 3 {(- 2)}^{2} e^{- 2}$

Step 10.3.2.1.5

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

$f' (- 2) = - \frac{8}{e^{2}} + 3 \cdot (4 e^{- 2})$

Step 10.3.2.1.6

Multiply $3$ by $4$ .

$f' (- 2) = - \frac{8}{e^{2}} + 12 e^{- 2}$

Step 10.3.2.1.7

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

$f' (- 2) = - \frac{8}{e^{2}} + 12 (\frac{1}{e^{2}})$

Step 10.3.2.1.8

Combine $12$ and $\frac{1}{e^{2}}$ .

$f' (- 2) = - \frac{8}{e^{2}} + \frac{12}{e^{2}}$

Step 10.3.2.2

Combine fractions.

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Step 10.3.2.2.1

Combine the numerators over the common denominator.

$f' (- 2) = \frac{- 8 + 12}{e^{2}}$

Step 10.3.2.2.2

Add $- 8$ and $12$ .

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

Step 10.3.2.3

The final answer is $\frac{4}{e^{2}}$ .

$\frac{4}{e^{2}}$

Step 10.4

Substitute any number, such as $2$ , from the interval $(0, \infty)$ in the first derivative $x^{3} e^{x} + 3 x^{2} e^{x}$ to check if the result is negative or positive.

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Step 10.4.1

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

$f' (2) = {(2)}^{3} e^{2} + 3 {(2)}^{2} e^{2}$

Step 10.4.2

Simplify the result.

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Step 10.4.2.1

Simplify each term.

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Step 10.4.2.1.1

Raise $2$ to the power of $3$ .

$f' (2) = 8 e^{2} + 3 {(2)}^{2} e^{2}$

Step 10.4.2.1.2

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

$f' (2) = 8 e^{2} + 3 \cdot (4 e^{2})$

Step 10.4.2.1.3

Multiply $3$ by $4$ .

$f' (2) = 8 e^{2} + 12 e^{2}$

Step 10.4.2.2

Add $8 e^{2}$ and $12 e^{2}$ .

$f' (2) = 20 e^{2}$

Step 10.4.2.3

The final answer is $20 e^{2}$ .

$20 e^{2}$

Step 10.5

Since the first derivative changed signs from negative to positive around $x = - 3$ , then $x = - 3$ is a local minimum.

$x = - 3$ is a local minimum

Step 10.6

Since the first derivative did not change signs around $x = 0$ , this is not a local maximum or minimum.

Not a local maximum or minimum

Step 10.7

These are the local extrema for $f (x) = x^{3} e^{x}$ .

$x = - 3$ is a local minimum

Step 11