Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

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

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

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

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

Step 1.3

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 $g (x) = 3 x$ .

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Step 1.3.1

To apply the Chain Rule, set $u$ as $3 x$ .

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

Step 1.3.2

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

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

Step 1.3.3

Replace all occurrences of $u$ with $3 x$ .

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

Step 1.4

Differentiate.

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Step 1.4.1

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

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

Step 1.4.2

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

$f' (x) = 2 (x (e^{3 x} (3 \cdot 1)) + e^{3 x} \frac{d}{d x} (x))$

Step 1.4.3

Simplify the expression.

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Step 1.4.3.1

Multiply $3$ by $1$ .

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

Step 1.4.3.2

Move $3$ to the left of $e^{3 x}$ .

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

Step 1.4.4

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

$f' (x) = 2 (x (3 e^{3 x}) + e^{3 x} \cdot 1)$

Step 1.4.5

Multiply $e^{3 x}$ by $1$ .

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

Step 1.5

Simplify.

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Step 1.5.1

Apply the distributive property.

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

Step 1.5.2

Multiply $3$ by $2$ .

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

Step 1.5.3

Reorder terms.

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

Step 1.5.4

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

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

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

Evaluate $\frac{d}{d x} [6 x e^{3 x}]$ .

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Step 2.2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x e^{3 x}$ with respect to $x$ is $6 \frac{d}{d x} [x e^{3 x}]$ .

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

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

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

Step 2.2.3

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 $g (x) = 3 x$ .

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Step 2.2.3.1

To apply the Chain Rule, set $u_{1}$ as $3 x$ .

$f'' (x) = 6 (x (\frac{d}{d u (1)} (e^{u_{1}}) \frac{d}{d x} (3 x)) + e^{3 x} \frac{d}{d x} (x)) + \frac{d}{d x} (2 e^{3 x})$

Step 2.2.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$f'' (x) = 6 (x (e^{u_{1}} \frac{d}{d x} (3 x)) + e^{3 x} \frac{d}{d x} (x)) + \frac{d}{d x} (2 e^{3 x})$

Step 2.2.3.3

Replace all occurrences of $u_{1}$ with $3 x$ .

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

Step 2.2.4

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

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

Step 2.2.5

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

$f'' (x) = 6 (x (e^{3 x} (3 \cdot 1)) + e^{3 x} \frac{d}{d x} (x)) + \frac{d}{d x} (2 e^{3 x})$

Step 2.2.6

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

$f'' (x) = 6 (x (e^{3 x} (3 \cdot 1)) + e^{3 x} \cdot 1) + \frac{d}{d x} (2 e^{3 x})$

Step 2.2.7

Multiply $3$ by $1$ .

$f'' (x) = 6 (x (e^{3 x} \cdot 3) + e^{3 x} \cdot 1) + \frac{d}{d x} (2 e^{3 x})$

Step 2.2.8

Move $3$ to the left of $e^{3 x}$ .

$f'' (x) = 6 (x (3 \cdot e^{3 x}) + e^{3 x} \cdot 1) + \frac{d}{d x} (2 e^{3 x})$

Step 2.2.9

Multiply $e^{3 x}$ by $1$ .

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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 $g (x) = 3 x$ .

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Step 2.3.2.1

To apply the Chain Rule, set $u_{2}$ as $3 x$ .

$f'' (x) = 6 (x (3 e^{3 x}) + e^{3 x}) + 2 (\frac{d}{d u (2)} (e^{u_{2}}) \frac{d}{d x} (3 x))$

Step 2.3.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

$f'' (x) = 6 (x (3 e^{3 x}) + e^{3 x}) + 2 (e^{u_{2}} \frac{d}{d x} (3 x))$

Step 2.3.2.3

Replace all occurrences of $u_{2}$ with $3 x$ .

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

Step 2.3.3

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

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

Step 2.3.4

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

$f'' (x) = 6 (x (3 e^{3 x}) + e^{3 x}) + 2 (e^{3 x} (3 \cdot 1))$

Step 2.3.5

Multiply $3$ by $1$ .

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

Step 2.3.6

Move $3$ to the left of $e^{3 x}$ .

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

Step 2.3.7

Multiply $3$ by $2$ .

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

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

Combine terms.

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Step 2.4.2.1

Multiply $3$ by $6$ .

$f'' (x) = 18 (x (e^{3 x})) + 6 e^{3 x} + 6 e^{3 x}$

Step 2.4.2.2

Add $6 e^{3 x}$ and $6 e^{3 x}$ .

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

Step 2.4.3

Reorder terms.

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

Step 2.4.4

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

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

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $18 x e^{3 x} + 12 e^{3 x}$ with respect to $x$ is $\frac{d}{d x} [18 x e^{3 x}] + \frac{d}{d x} [12 e^{3 x}]$ .

$f''' (x) = \frac{d}{d x} (18 x e^{3 x}) + \frac{d}{d x} (12 e^{3 x})$

Step 3.2

Evaluate $\frac{d}{d x} [18 x e^{3 x}]$ .

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Step 3.2.1

Since $18$ is constant with respect to $x$ , the derivative of $18 x e^{3 x}$ with respect to $x$ is $18 \frac{d}{d x} [x e^{3 x}]$ .

$f''' (x) = 18 \frac{d}{d x} (x e^{3 x}) + \frac{d}{d x} (12 e^{3 x})$

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

$f''' (x) = 18 (x \frac{d}{d x} (e^{3 x}) + e^{3 x} \frac{d}{d x} (x)) + \frac{d}{d x} (12 e^{3 x})$

Step 3.2.3

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 $g (x) = 3 x$ .

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Step 3.2.3.1

To apply the Chain Rule, set $u_{1}$ as $3 x$ .

$f''' (x) = 18 (x (\frac{d}{d u (1)} (e^{u_{1}}) \frac{d}{d x} (3 x)) + e^{3 x} \frac{d}{d x} (x)) + \frac{d}{d x} (12 e^{3 x})$

Step 3.2.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$f''' (x) = 18 (x (e^{u_{1}} \frac{d}{d x} (3 x)) + e^{3 x} \frac{d}{d x} (x)) + \frac{d}{d x} (12 e^{3 x})$

Step 3.2.3.3

Replace all occurrences of $u_{1}$ with $3 x$ .

$f''' (x) = 18 (x (e^{3 x} \frac{d}{d x} (3 x)) + e^{3 x} \frac{d}{d x} (x)) + \frac{d}{d x} (12 e^{3 x})$

Step 3.2.4

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

$f''' (x) = 18 (x (e^{3 x} (3 \frac{d}{d x} (x))) + e^{3 x} \frac{d}{d x} (x)) + \frac{d}{d x} (12 e^{3 x})$

Step 3.2.5

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

$f''' (x) = 18 (x (e^{3 x} (3 \cdot 1)) + e^{3 x} \frac{d}{d x} (x)) + \frac{d}{d x} (12 e^{3 x})$

Step 3.2.6

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

$f''' (x) = 18 (x (e^{3 x} (3 \cdot 1)) + e^{3 x} \cdot 1) + \frac{d}{d x} (12 e^{3 x})$

Step 3.2.7

Multiply $3$ by $1$ .

$f''' (x) = 18 (x (e^{3 x} \cdot 3) + e^{3 x} \cdot 1) + \frac{d}{d x} (12 e^{3 x})$

Step 3.2.8

Move $3$ to the left of $e^{3 x}$ .

$f''' (x) = 18 (x (3 \cdot e^{3 x}) + e^{3 x} \cdot 1) + \frac{d}{d x} (12 e^{3 x})$

Step 3.2.9

Multiply $e^{3 x}$ by $1$ .

$f''' (x) = 18 (x (3 e^{3 x}) + e^{3 x}) + \frac{d}{d x} (12 e^{3 x})$

Step 3.3

Evaluate $\frac{d}{d x} [12 e^{3 x}]$ .

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Step 3.3.1

Since $12$ is constant with respect to $x$ , the derivative of $12 e^{3 x}$ with respect to $x$ is $12 \frac{d}{d x} [e^{3 x}]$ .

$f''' (x) = 18 (x (3 e^{3 x}) + e^{3 x}) + 12 \frac{d}{d x} (e^{3 x})$

Step 3.3.2

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 $g (x) = 3 x$ .

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Step 3.3.2.1

To apply the Chain Rule, set $u_{2}$ as $3 x$ .

$f''' (x) = 18 (x (3 e^{3 x}) + e^{3 x}) + 12 (\frac{d}{d u (2)} (e^{u_{2}}) \frac{d}{d x} (3 x))$

Step 3.3.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

$f''' (x) = 18 (x (3 e^{3 x}) + e^{3 x}) + 12 (e^{u_{2}} \frac{d}{d x} (3 x))$

Step 3.3.2.3

Replace all occurrences of $u_{2}$ with $3 x$ .

$f''' (x) = 18 (x (3 e^{3 x}) + e^{3 x}) + 12 (e^{3 x} \frac{d}{d x} (3 x))$

Step 3.3.3

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

$f''' (x) = 18 (x (3 e^{3 x}) + e^{3 x}) + 12 (e^{3 x} (3 \frac{d}{d x} (x)))$

Step 3.3.4

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

$f''' (x) = 18 (x (3 e^{3 x}) + e^{3 x}) + 12 (e^{3 x} (3 \cdot 1))$

Step 3.3.5

Multiply $3$ by $1$ .

$f''' (x) = 18 (x (3 e^{3 x}) + e^{3 x}) + 12 (e^{3 x} \cdot 3)$

Step 3.3.6

Move $3$ to the left of $e^{3 x}$ .

$f''' (x) = 18 (x (3 e^{3 x}) + e^{3 x}) + 12 (3 \cdot e^{3 x})$

Step 3.3.7

Multiply $3$ by $12$ .

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

Step 3.4

Simplify.

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Step 3.4.1

Apply the distributive property.

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

Step 3.4.2

Combine terms.

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Step 3.4.2.1

Multiply $3$ by $18$ .

$f''' (x) = 54 (x (e^{3 x})) + 18 e^{3 x} + 36 e^{3 x}$

Step 3.4.2.2

Add $18 e^{3 x}$ and $36 e^{3 x}$ .

$f''' (x) = 54 x e^{3 x} + 54 e^{3 x}$

Step 3.4.3

Reorder terms.

$f''' (x) = 54 e^{3 x} x + 54 e^{3 x}$

Step 3.4.4

Reorder factors in $54 e^{3 x} x + 54 e^{3 x}$ .

$f''' (x) = 54 x e^{3 x} + 54 e^{3 x}$

Step 4

Find the fourth derivative.

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Step 4.1

By the Sum Rule, the derivative of $54 x e^{3 x} + 54 e^{3 x}$ with respect to $x$ is $\frac{d}{d x} [54 x e^{3 x}] + \frac{d}{d x} [54 e^{3 x}]$ .

$f^{4} (x) = \frac{d}{d x} (54 x e^{3 x}) + \frac{d}{d x} (54 e^{3 x})$

Step 4.2

Evaluate $\frac{d}{d x} [54 x e^{3 x}]$ .

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Step 4.2.1

Since $54$ is constant with respect to $x$ , the derivative of $54 x e^{3 x}$ with respect to $x$ is $54 \frac{d}{d x} [x e^{3 x}]$ .

$f^{4} (x) = 54 \frac{d}{d x} (x e^{3 x}) + \frac{d}{d x} (54 e^{3 x})$

Step 4.2.2

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

$f^{4} (x) = 54 (x \frac{d}{d x} (e^{3 x}) + e^{3 x} \frac{d}{d x} (x)) + \frac{d}{d x} (54 e^{3 x})$

Step 4.2.3

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 $g (x) = 3 x$ .

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Step 4.2.3.1

To apply the Chain Rule, set $u_{1}$ as $3 x$ .

$f^{4} (x) = 54 (x (\frac{d}{d u (1)} (e^{u_{1}}) \frac{d}{d x} (3 x)) + e^{3 x} \frac{d}{d x} (x)) + \frac{d}{d x} (54 e^{3 x})$

Step 4.2.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$f^{4} (x) = 54 (x (e^{u_{1}} \frac{d}{d x} (3 x)) + e^{3 x} \frac{d}{d x} (x)) + \frac{d}{d x} (54 e^{3 x})$

Step 4.2.3.3

Replace all occurrences of $u_{1}$ with $3 x$ .

$f^{4} (x) = 54 (x (e^{3 x} \frac{d}{d x} (3 x)) + e^{3 x} \frac{d}{d x} (x)) + \frac{d}{d x} (54 e^{3 x})$

Step 4.2.4

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

$f^{4} (x) = 54 (x (e^{3 x} (3 \frac{d}{d x} (x))) + e^{3 x} \frac{d}{d x} (x)) + \frac{d}{d x} (54 e^{3 x})$

Step 4.2.5

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

$f^{4} (x) = 54 (x (e^{3 x} (3 \cdot 1)) + e^{3 x} \frac{d}{d x} (x)) + \frac{d}{d x} (54 e^{3 x})$

Step 4.2.6

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

$f^{4} (x) = 54 (x (e^{3 x} (3 \cdot 1)) + e^{3 x} \cdot 1) + \frac{d}{d x} (54 e^{3 x})$

Step 4.2.7

Multiply $3$ by $1$ .

$f^{4} (x) = 54 (x (e^{3 x} \cdot 3) + e^{3 x} \cdot 1) + \frac{d}{d x} (54 e^{3 x})$

Step 4.2.8

Move $3$ to the left of $e^{3 x}$ .

$f^{4} (x) = 54 (x (3 \cdot e^{3 x}) + e^{3 x} \cdot 1) + \frac{d}{d x} (54 e^{3 x})$

Step 4.2.9

Multiply $e^{3 x}$ by $1$ .

$f^{4} (x) = 54 (x (3 e^{3 x}) + e^{3 x}) + \frac{d}{d x} (54 e^{3 x})$

Step 4.3

Evaluate $\frac{d}{d x} [54 e^{3 x}]$ .

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Step 4.3.1

Since $54$ is constant with respect to $x$ , the derivative of $54 e^{3 x}$ with respect to $x$ is $54 \frac{d}{d x} [e^{3 x}]$ .

$f^{4} (x) = 54 (x (3 e^{3 x}) + e^{3 x}) + 54 \frac{d}{d x} (e^{3 x})$

Step 4.3.2

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 $g (x) = 3 x$ .

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Step 4.3.2.1

To apply the Chain Rule, set $u_{2}$ as $3 x$ .

$f^{4} (x) = 54 (x (3 e^{3 x}) + e^{3 x}) + 54 (\frac{d}{d u (2)} (e^{u_{2}}) \frac{d}{d x} (3 x))$

Step 4.3.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

$f^{4} (x) = 54 (x (3 e^{3 x}) + e^{3 x}) + 54 (e^{u_{2}} \frac{d}{d x} (3 x))$

Step 4.3.2.3

Replace all occurrences of $u_{2}$ with $3 x$ .

$f^{4} (x) = 54 (x (3 e^{3 x}) + e^{3 x}) + 54 (e^{3 x} \frac{d}{d x} (3 x))$

Step 4.3.3

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

$f^{4} (x) = 54 (x (3 e^{3 x}) + e^{3 x}) + 54 (e^{3 x} (3 \frac{d}{d x} (x)))$

Step 4.3.4

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

$f^{4} (x) = 54 (x (3 e^{3 x}) + e^{3 x}) + 54 (e^{3 x} (3 \cdot 1))$

Step 4.3.5

Multiply $3$ by $1$ .

$f^{4} (x) = 54 (x (3 e^{3 x}) + e^{3 x}) + 54 (e^{3 x} \cdot 3)$

Step 4.3.6

Move $3$ to the left of $e^{3 x}$ .

$f^{4} (x) = 54 (x (3 e^{3 x}) + e^{3 x}) + 54 (3 \cdot e^{3 x})$

Step 4.3.7

Multiply $3$ by $54$ .

$f^{4} (x) = 54 (x (3 e^{3 x}) + e^{3 x}) + 162 e^{3 x}$

Step 4.4

Simplify.

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Step 4.4.1

Apply the distributive property.

$f^{4} (x) = 54 (x (3 e^{3 x})) + 54 e^{3 x} + 162 e^{3 x}$

Step 4.4.2

Combine terms.

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Step 4.4.2.1

Multiply $3$ by $54$ .

$f^{4} (x) = 162 (x (e^{3 x})) + 54 e^{3 x} + 162 e^{3 x}$

Step 4.4.2.2

Add $54 e^{3 x}$ and $162 e^{3 x}$ .

$f^{4} (x) = 162 x e^{3 x} + 216 e^{3 x}$

Step 4.4.3

Reorder terms.

$f^{4} (x) = 162 e^{3 x} x + 216 e^{3 x}$

Step 4.4.4

Reorder factors in $162 e^{3 x} x + 216 e^{3 x}$ .

$f^{4} (x) = 162 x e^{3 x} + 216 e^{3 x}$

Step 5

The fourth derivative of $f (x)$ with respect to $x$ is $162 x e^{3 x} + 216 e^{3 x}$ .

$162 x e^{3 x} + 216 e^{3 x}$