Calculus Examples

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

Step 1

Find the first derivative.

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Simplify.

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Step 1.4.1

Reorder terms.

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

Step 1.4.2

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

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

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

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

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.3

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

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Step 2.3.1

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

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

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

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

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

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

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

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Step 2.4.2.1

Move $e^{x}$ .

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

Step 2.4.2.2

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

$x^{2} e^{x} + 4 x e^{x} + 2 e^{x}$

Step 2.4.3

Reorder terms.

$e^{x} x^{2} + 4 e^{x} x + 2 e^{x}$

Step 2.4.4

Reorder factors in $e^{x} x^{2} + 4 e^{x} x + 2 e^{x}$ .

$f'' (x) = x^{2} e^{x} + 4 x e^{x} + 2 e^{x}$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $x^{2} e^{x} + 4 x e^{x} + 2 e^{x}$ with respect to $x$ is $\frac{d}{d x} [x^{2} e^{x}] + \frac{d}{d x} [4 x e^{x}] + \frac{d}{d x} [2 e^{x}]$ .

$\frac{d}{d x} [x^{2} e^{x}] + \frac{d}{d x} [4 x e^{x}] + \frac{d}{d x} [2 e^{x}]$

Step 3.2

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

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Step 3.2.1

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

$x^{2} \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [4 x e^{x}] + \frac{d}{d x} [2 e^{x}]$

Step 3.2.2

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

$x^{2} e^{x} + e^{x} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [4 x e^{x}] + \frac{d}{d x} [2 e^{x}]$

Step 3.2.3

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

$x^{2} e^{x} + e^{x} (2 x) + \frac{d}{d x} [4 x e^{x}] + \frac{d}{d x} [2 e^{x}]$

Step 3.3

Evaluate $\frac{d}{d x} [4 x e^{x}]$ .

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Step 3.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x e^{x}$ with respect to $x$ is $4 \frac{d}{d x} [x e^{x}]$ .

$x^{2} e^{x} + e^{x} (2 x) + 4 \frac{d}{d x} [x e^{x}] + \frac{d}{d x} [2 e^{x}]$

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

$x^{2} e^{x} + e^{x} (2 x) + 4 (x \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x]) + \frac{d}{d x} [2 e^{x}]$

Step 3.3.3

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

$x^{2} e^{x} + e^{x} (2 x) + 4 (x e^{x} + e^{x} \frac{d}{d x} [x]) + \frac{d}{d x} [2 e^{x}]$

Step 3.3.4

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

$x^{2} e^{x} + e^{x} (2 x) + 4 (x e^{x} + e^{x} \cdot 1) + \frac{d}{d x} [2 e^{x}]$

Step 3.3.5

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

$x^{2} e^{x} + e^{x} (2 x) + 4 (x e^{x} + e^{x}) + \frac{d}{d x} [2 e^{x}]$

Step 3.4

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

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Step 3.4.1

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

$x^{2} e^{x} + e^{x} (2 x) + 4 (x e^{x} + e^{x}) + 2 \frac{d}{d x} [e^{x}]$

Step 3.4.2

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

$x^{2} e^{x} + e^{x} (2 x) + 4 (x e^{x} + e^{x}) + 2 e^{x}$

Step 3.5

Simplify.

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Step 3.5.1

Apply the distributive property.

$x^{2} e^{x} + e^{x} (2 x) + 4 (x e^{x}) + 4 e^{x} + 2 e^{x}$

Step 3.5.2

Combine terms.

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Step 3.5.2.1

Add $e^{x} (2 x)$ and $4 x e^{x}$ .

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Step 3.5.2.1.1

Move $e^{x}$ .

$x^{2} e^{x} + 2 x e^{x} + 4 x e^{x} + 4 e^{x} + 2 e^{x}$

Step 3.5.2.1.2

Add $2 x e^{x}$ and $4 x e^{x}$ .

$x^{2} e^{x} + 6 x e^{x} + 4 e^{x} + 2 e^{x}$

Step 3.5.2.2

Add $4 e^{x}$ and $2 e^{x}$ .

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

Step 3.5.3

Reorder terms.

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

Step 3.5.4

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

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

Step 4

Find the fourth derivative.

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.2.3

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

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

Step 4.3

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

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Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.3.3

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

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

Step 4.3.4

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

$x^{2} e^{x} + e^{x} (2 x) + 6 (x e^{x} + e^{x} \cdot 1) + \frac{d}{d x} [6 e^{x}]$

Step 4.3.5

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

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

Step 4.4

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

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Step 4.4.1

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

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

Step 4.4.2

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

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

Step 4.5

Simplify.

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Step 4.5.1

Apply the distributive property.

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

Step 4.5.2

Combine terms.

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Step 4.5.2.1

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

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Step 4.5.2.1.1

Move $e^{x}$ .

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

Step 4.5.2.1.2

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

$x^{2} e^{x} + 8 x e^{x} + 6 e^{x} + 6 e^{x}$

Step 4.5.2.2

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

$x^{2} e^{x} + 8 x e^{x} + 12 e^{x}$

Step 4.5.3

Reorder terms.

$e^{x} x^{2} + 8 e^{x} x + 12 e^{x}$

Step 4.5.4

Reorder factors in $e^{x} x^{2} + 8 e^{x} x + 12 e^{x}$ .

$f^{4} (x) = x^{2} e^{x} + 8 x e^{x} + 12 e^{x}$