Calculus Examples

$y = 2 x^{3} e^{x}$

Step 1

Find the first derivative.

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Step 1.1

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

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

Step 1.2

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

$2 (x^{3} \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{3}])$

Step 1.3

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

$2 (x^{3} e^{x} + e^{x} \frac{d}{d x} [x^{3}])$

Step 1.4

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

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

Step 1.5

Simplify.

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Step 1.5.1

Apply the distributive property.

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

Step 1.5.2

Multiply $3$ by $2$ .

$2 x^{3} e^{x} + 6 e^{x} x^{2}$

Step 1.5.3

Reorder terms.

$2 e^{x} x^{3} + 6 e^{x} x^{2}$

Step 1.5.4

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

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

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $2 x^{3} e^{x} + 6 x^{2} e^{x}$ with respect to $x$ is $\frac{d}{d x} [2 x^{3} e^{x}] + \frac{d}{d x} [6 x^{2} e^{x}]$ .

$\frac{d}{d x} [2 x^{3} e^{x}] + \frac{d}{d x} [6 x^{2} e^{x}]$

Step 2.2

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

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Step 2.2.1

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

$2 \frac{d}{d x} [x^{3} e^{x}] + \frac{d}{d x} [6 x^{2} e^{x}]$

Step 2.2.2

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

$2 (x^{3} \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{3}]) + \frac{d}{d x} [6 x^{2} e^{x}]$

Step 2.2.3

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

$2 (x^{3} e^{x} + e^{x} \frac{d}{d x} [x^{3}]) + \frac{d}{d x} [6 x^{2} e^{x}]$

Step 2.2.4

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

$2 (x^{3} e^{x} + e^{x} (3 x^{2})) + \frac{d}{d x} [6 x^{2} e^{x}]$

Step 2.3

Evaluate $\frac{d}{d x} [6 x^{2} e^{x}]$ .

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Step 2.3.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{2} e^{x}$ with respect to $x$ is $6 \frac{d}{d x} [x^{2} e^{x}]$ .

$2 (x^{3} e^{x} + e^{x} (3 x^{2})) + 6 \frac{d}{d x} [x^{2} e^{x}]$

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

$2 (x^{3} e^{x} + e^{x} (3 x^{2})) + 6 (x^{2} \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{2}])$

Step 2.3.3

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

$2 (x^{3} e^{x} + e^{x} (3 x^{2})) + 6 (x^{2} e^{x} + e^{x} \frac{d}{d x} [x^{2}])$

Step 2.3.4

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

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

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

Apply the distributive property.

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

Step 2.4.3

Combine terms.

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Step 2.4.3.1

Multiply $3$ by $2$ .

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

Step 2.4.3.2

Multiply $2$ by $6$ .

$2 x^{3} e^{x} + 6 e^{x} x^{2} + 6 x^{2} e^{x} + 12 (e^{x} (x))$

Step 2.4.3.3

Add $6 e^{x} x^{2}$ and $6 x^{2} e^{x}$ .

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Step 2.4.3.3.1

Move $e^{x}$ .

$2 x^{3} e^{x} + 6 x^{2} e^{x} + 6 x^{2} e^{x} + 12 e^{x} x$

Step 2.4.3.3.2

Add $6 x^{2} e^{x}$ and $6 x^{2} e^{x}$ .

$2 x^{3} e^{x} + 12 x^{2} e^{x} + 12 e^{x} x$

Step 2.4.4

Reorder terms.

$2 e^{x} x^{3} + 12 e^{x} x^{2} + 12 e^{x} x$

Step 2.4.5

Reorder factors in $2 e^{x} x^{3} + 12 e^{x} x^{2} + 12 e^{x} x$ .

$f'' (x) = 2 x^{3} e^{x} + 12 x^{2} e^{x} + 12 x e^{x}$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $2 x^{3} e^{x} + 12 x^{2} e^{x} + 12 x e^{x}$ with respect to $x$ is $\frac{d}{d x} [2 x^{3} e^{x}] + \frac{d}{d x} [12 x^{2} e^{x}] + \frac{d}{d x} [12 x e^{x}]$ .

$\frac{d}{d x} [2 x^{3} e^{x}] + \frac{d}{d x} [12 x^{2} e^{x}] + \frac{d}{d x} [12 x e^{x}]$

Step 3.2

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

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Step 3.2.1

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

$2 \frac{d}{d x} [x^{3} e^{x}] + \frac{d}{d x} [12 x^{2} e^{x}] + \frac{d}{d x} [12 x e^{x}]$

Step 3.2.2

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

$2 (x^{3} \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{3}]) + \frac{d}{d x} [12 x^{2} e^{x}] + \frac{d}{d x} [12 x e^{x}]$

Step 3.2.3

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

$2 (x^{3} e^{x} + e^{x} \frac{d}{d x} [x^{3}]) + \frac{d}{d x} [12 x^{2} e^{x}] + \frac{d}{d x} [12 x e^{x}]$

Step 3.2.4

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

$2 (x^{3} e^{x} + e^{x} (3 x^{2})) + \frac{d}{d x} [12 x^{2} e^{x}] + \frac{d}{d x} [12 x e^{x}]$

Step 3.3

Evaluate $\frac{d}{d x} [12 x^{2} e^{x}]$ .

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Step 3.3.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{2} e^{x}$ with respect to $x$ is $12 \frac{d}{d x} [x^{2} e^{x}]$ .

$2 (x^{3} e^{x} + e^{x} (3 x^{2})) + 12 \frac{d}{d x} [x^{2} e^{x}] + \frac{d}{d x} [12 x e^{x}]$

Step 3.3.2

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

$2 (x^{3} e^{x} + e^{x} (3 x^{2})) + 12 (x^{2} \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [12 x e^{x}]$

Step 3.3.3

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

$2 (x^{3} e^{x} + e^{x} (3 x^{2})) + 12 (x^{2} e^{x} + e^{x} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [12 x e^{x}]$

Step 3.3.4

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

$2 (x^{3} e^{x} + e^{x} (3 x^{2})) + 12 (x^{2} e^{x} + e^{x} (2 x)) + \frac{d}{d x} [12 x e^{x}]$

Step 3.4

Evaluate $\frac{d}{d x} [12 x e^{x}]$ .

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Step 3.4.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x e^{x}$ with respect to $x$ is $12 \frac{d}{d x} [x e^{x}]$ .

$2 (x^{3} e^{x} + e^{x} (3 x^{2})) + 12 (x^{2} e^{x} + e^{x} (2 x)) + 12 \frac{d}{d x} [x e^{x}]$

Step 3.4.2

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

$2 (x^{3} e^{x} + e^{x} (3 x^{2})) + 12 (x^{2} e^{x} + e^{x} (2 x)) + 12 (x \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x])$

Step 3.4.3

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

$2 (x^{3} e^{x} + e^{x} (3 x^{2})) + 12 (x^{2} e^{x} + e^{x} (2 x)) + 12 (x e^{x} + e^{x} \frac{d}{d x} [x])$

Step 3.4.4

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

$2 (x^{3} e^{x} + e^{x} (3 x^{2})) + 12 (x^{2} e^{x} + e^{x} (2 x)) + 12 (x e^{x} + e^{x} \cdot 1)$

Step 3.4.5

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

$2 (x^{3} e^{x} + e^{x} (3 x^{2})) + 12 (x^{2} e^{x} + e^{x} (2 x)) + 12 (x e^{x} + e^{x})$

Step 3.5

Simplify.

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Step 3.5.1

Apply the distributive property.

$2 (x^{3} e^{x}) + 2 (e^{x} (3 x^{2})) + 12 (x^{2} e^{x} + e^{x} (2 x)) + 12 (x e^{x} + e^{x})$

Step 3.5.2

Apply the distributive property.

$2 (x^{3} e^{x}) + 2 (e^{x} (3 x^{2})) + 12 (x^{2} e^{x}) + 12 (e^{x} (2 x)) + 12 (x e^{x} + e^{x})$

Step 3.5.3

Apply the distributive property.

$2 (x^{3} e^{x}) + 2 (e^{x} (3 x^{2})) + 12 (x^{2} e^{x}) + 12 (e^{x} (2 x)) + 12 (x e^{x}) + 12 e^{x}$

Step 3.5.4

Combine terms.

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Step 3.5.4.1

Multiply $3$ by $2$ .

$2 x^{3} e^{x} + 6 (e^{x} (x^{2})) + 12 (x^{2} e^{x}) + 12 (e^{x} (2 x)) + 12 (x e^{x}) + 12 e^{x}$

Step 3.5.4.2

Multiply $2$ by $12$ .

$2 x^{3} e^{x} + 6 e^{x} x^{2} + 12 x^{2} e^{x} + 24 (e^{x} (x)) + 12 (x e^{x}) + 12 e^{x}$

Step 3.5.4.3

Add $6 e^{x} x^{2}$ and $12 x^{2} e^{x}$ .

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Step 3.5.4.3.1

Move $e^{x}$ .

$2 x^{3} e^{x} + 6 x^{2} e^{x} + 12 x^{2} e^{x} + 24 e^{x} x + 12 (x e^{x}) + 12 e^{x}$

Step 3.5.4.3.2

Add $6 x^{2} e^{x}$ and $12 x^{2} e^{x}$ .

$2 x^{3} e^{x} + 18 x^{2} e^{x} + 24 e^{x} x + 12 (x e^{x}) + 12 e^{x}$

Step 3.5.4.4

Add $24 e^{x} x$ and $12 x e^{x}$ .

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Step 3.5.4.4.1

Move $e^{x}$ .

$2 x^{3} e^{x} + 18 x^{2} e^{x} + 24 x e^{x} + 12 x e^{x} + 12 e^{x}$

Step 3.5.4.4.2

Add $24 x e^{x}$ and $12 x e^{x}$ .

$2 x^{3} e^{x} + 18 x^{2} e^{x} + 36 x e^{x} + 12 e^{x}$

Step 3.5.5

Reorder terms.

$2 e^{x} x^{3} + 18 e^{x} x^{2} + 36 e^{x} x + 12 e^{x}$

Step 3.5.6

Reorder factors in $2 e^{x} x^{3} + 18 e^{x} x^{2} + 36 e^{x} x + 12 e^{x}$ .

$f''' (x) = 2 x^{3} e^{x} + 18 x^{2} e^{x} + 36 x e^{x} + 12 e^{x}$