Calculus Examples

$f (x) = e^{9 x} + 7 \cos (x) \sin (x)$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $e^{9 x} + 7 \cos (x) \sin (x)$ with respect to $x$ is $\frac{d}{d x} [e^{9 x}] + \frac{d}{d x} [7 \cos (x) \sin (x)]$ .

$\frac{d}{d x} [e^{9 x}] + \frac{d}{d x} [7 \cos (x) \sin (x)]$

Step 1.2

Evaluate $\frac{d}{d x} [e^{9 x}]$ .

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Step 1.2.1

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 $g (x) = 9 x$ .

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Step 1.2.1.1

To apply the Chain Rule, set $u$ as $9 x$ .

$\frac{d}{d u} [e^{u}] \frac{d}{d x} [9 x] + \frac{d}{d x} [7 \cos (x) \sin (x)]$

Step 1.2.1.2

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

$e^{u} \frac{d}{d x} [9 x] + \frac{d}{d x} [7 \cos (x) \sin (x)]$

Step 1.2.1.3

Replace all occurrences of $u$ with $9 x$ .

$e^{9 x} \frac{d}{d x} [9 x] + \frac{d}{d x} [7 \cos (x) \sin (x)]$

Step 1.2.2

Since $9$ is constant with respect to $x$ , the derivative of $9 x$ with respect to $x$ is $9 \frac{d}{d x} [x]$ .

$e^{9 x} (9 \frac{d}{d x} [x]) + \frac{d}{d x} [7 \cos (x) \sin (x)]$

Step 1.2.3

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

$e^{9 x} (9 \cdot 1) + \frac{d}{d x} [7 \cos (x) \sin (x)]$

Step 1.2.4

Multiply $9$ by $1$ .

$e^{9 x} \cdot 9 + \frac{d}{d x} [7 \cos (x) \sin (x)]$

Step 1.2.5

Move $9$ to the left of $e^{9 x}$ .

$9 e^{9 x} + \frac{d}{d x} [7 \cos (x) \sin (x)]$

Step 1.3

Evaluate $\frac{d}{d x} [7 \cos (x) \sin (x)]$ .

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Step 1.3.1

Since $7$ is constant with respect to $x$ , the derivative of $7 \cos (x) \sin (x)$ with respect to $x$ is $7 \frac{d}{d x} [\cos (x) \sin (x)]$ .

$9 e^{9 x} + 7 \frac{d}{d x} [\cos (x) \sin (x)]$

Step 1.3.2

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 $f (x) = \cos (x)$ and $g (x) = \sin (x)$ .

$9 e^{9 x} + 7 (\cos (x) \frac{d}{d x} [\sin (x)] + \sin (x) \frac{d}{d x} [\cos (x)])$

Step 1.3.3

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$9 e^{9 x} + 7 (\cos (x) \cos (x) + \sin (x) \frac{d}{d x} [\cos (x)])$

Step 1.3.4

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$9 e^{9 x} + 7 (\cos (x) \cos (x) + \sin (x) (- \sin (x)))$

Step 1.3.5

Raise $\cos (x)$ to the power of $1$ .

$9 e^{9 x} + 7 (\cos^{1} (x) \cos (x) + \sin (x) (- \sin (x)))$

Step 1.3.6

Raise $\cos (x)$ to the power of $1$ .

$9 e^{9 x} + 7 (\cos^{1} (x) \cos^{1} (x) + \sin (x) (- \sin (x)))$

Step 1.3.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$9 e^{9 x} + 7 ({\cos (x)}^{1 + 1} + \sin (x) (- \sin (x)))$

Step 1.3.8

Add $1$ and $1$ .

$9 e^{9 x} + 7 (\cos^{2} (x) + \sin (x) (- \sin (x)))$

Step 1.3.9

Raise $\sin (x)$ to the power of $1$ .

$9 e^{9 x} + 7 (\cos^{2} (x) - (\sin^{1} (x) \sin (x)))$

Step 1.3.10

Raise $\sin (x)$ to the power of $1$ .

$9 e^{9 x} + 7 (\cos^{2} (x) - (\sin^{1} (x) \sin^{1} (x)))$

Step 1.3.11

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$9 e^{9 x} + 7 (\cos^{2} (x) - {\sin (x)}^{1 + 1})$

Step 1.3.12

Add $1$ and $1$ .

$9 e^{9 x} + 7 (\cos^{2} (x) - \sin^{2} (x))$

Step 1.4

Simplify.

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Step 1.4.1

Apply the distributive property.

$9 e^{9 x} + 7 \cos^{2} (x) + 7 (- \sin^{2} (x))$

Step 1.4.2

Multiply $- 1$ by $7$ .

$f' (x) = 9 e^{9 x} + 7 \cos^{2} (x) - 7 \sin^{2} (x)$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $9 e^{9 x} + 7 \cos^{2} (x) - 7 \sin^{2} (x)$ with respect to $x$ is $\frac{d}{d x} [9 e^{9 x}] + \frac{d}{d x} [7 \cos^{2} (x)] + \frac{d}{d x} [- 7 \sin^{2} (x)]$ .

$\frac{d}{d x} [9 e^{9 x}] + \frac{d}{d x} [7 \cos^{2} (x)] + \frac{d}{d x} [- 7 \sin^{2} (x)]$

Step 2.2

Evaluate $\frac{d}{d x} [9 e^{9 x}]$ .

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Step 2.2.1

Since $9$ is constant with respect to $x$ , the derivative of $9 e^{9 x}$ with respect to $x$ is $9 \frac{d}{d x} [e^{9 x}]$ .

$9 \frac{d}{d x} [e^{9 x}] + \frac{d}{d x} [7 \cos^{2} (x)] + \frac{d}{d x} [- 7 \sin^{2} (x)]$

Step 2.2.2

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 $g (x) = 9 x$ .

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Step 2.2.2.1

To apply the Chain Rule, set $u_{1}$ as $9 x$ .

$9 (\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [9 x]) + \frac{d}{d x} [7 \cos^{2} (x)] + \frac{d}{d x} [- 7 \sin^{2} (x)]$

Step 2.2.2.2

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

$9 (e^{u_{1}} \frac{d}{d x} [9 x]) + \frac{d}{d x} [7 \cos^{2} (x)] + \frac{d}{d x} [- 7 \sin^{2} (x)]$

Step 2.2.2.3

Replace all occurrences of $u_{1}$ with $9 x$ .

$9 (e^{9 x} \frac{d}{d x} [9 x]) + \frac{d}{d x} [7 \cos^{2} (x)] + \frac{d}{d x} [- 7 \sin^{2} (x)]$

Step 2.2.3

Since $9$ is constant with respect to $x$ , the derivative of $9 x$ with respect to $x$ is $9 \frac{d}{d x} [x]$ .

$9 (e^{9 x} (9 \frac{d}{d x} [x])) + \frac{d}{d x} [7 \cos^{2} (x)] + \frac{d}{d x} [- 7 \sin^{2} (x)]$

Step 2.2.4

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

$9 (e^{9 x} (9 \cdot 1)) + \frac{d}{d x} [7 \cos^{2} (x)] + \frac{d}{d x} [- 7 \sin^{2} (x)]$

Step 2.2.5

Multiply $9$ by $1$ .

$9 (e^{9 x} \cdot 9) + \frac{d}{d x} [7 \cos^{2} (x)] + \frac{d}{d x} [- 7 \sin^{2} (x)]$

Step 2.2.6

Move $9$ to the left of $e^{9 x}$ .

$9 (9 \cdot e^{9 x}) + \frac{d}{d x} [7 \cos^{2} (x)] + \frac{d}{d x} [- 7 \sin^{2} (x)]$

Step 2.2.7

Multiply $9$ by $9$ .

$81 e^{9 x} + \frac{d}{d x} [7 \cos^{2} (x)] + \frac{d}{d x} [- 7 \sin^{2} (x)]$

Step 2.3

Evaluate $\frac{d}{d x} [7 \cos^{2} (x)]$ .

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Step 2.3.1

Since $7$ is constant with respect to $x$ , the derivative of $7 \cos^{2} (x)$ with respect to $x$ is $7 \frac{d}{d x} [\cos^{2} (x)]$ .

$81 e^{9 x} + 7 \frac{d}{d x} [\cos^{2} (x)] + \frac{d}{d x} [- 7 \sin^{2} (x)]$

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

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Step 2.3.2.1

To apply the Chain Rule, set $u_{2}$ as $\cos (x)$ .

$81 e^{9 x} + 7 (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [\cos (x)]) + \frac{d}{d x} [- 7 \sin^{2} (x)]$

Step 2.3.2.2

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

$81 e^{9 x} + 7 (2 u_{2} \frac{d}{d x} [\cos (x)]) + \frac{d}{d x} [- 7 \sin^{2} (x)]$

Step 2.3.2.3

Replace all occurrences of $u_{2}$ with $\cos (x)$ .

$81 e^{9 x} + 7 (2 \cos (x) \frac{d}{d x} [\cos (x)]) + \frac{d}{d x} [- 7 \sin^{2} (x)]$

Step 2.3.3

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$81 e^{9 x} + 7 (2 \cos (x) (- \sin (x))) + \frac{d}{d x} [- 7 \sin^{2} (x)]$

Step 2.3.4

Multiply $- 1$ by $2$ .

$81 e^{9 x} + 7 (- 2 \cos (x) \sin (x)) + \frac{d}{d x} [- 7 \sin^{2} (x)]$

Step 2.3.5

Multiply $- 2$ by $7$ .

$81 e^{9 x} - 14 \cos (x) \sin (x) + \frac{d}{d x} [- 7 \sin^{2} (x)]$

Step 2.4

Evaluate $\frac{d}{d x} [- 7 \sin^{2} (x)]$ .

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Step 2.4.1

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 \sin^{2} (x)$ with respect to $x$ is $- 7 \frac{d}{d x} [\sin^{2} (x)]$ .

$81 e^{9 x} - 14 \cos (x) \sin (x) - 7 \frac{d}{d x} [\sin^{2} (x)]$

Step 2.4.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \sin (x)$ .

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Step 2.4.2.1

To apply the Chain Rule, set $u_{3}$ as $\sin (x)$ .

$81 e^{9 x} - 14 \cos (x) \sin (x) - 7 (\frac{d}{d u_{3}} [{u_{3}}^{2}] \frac{d}{d x} [\sin (x)])$

Step 2.4.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{3}} [{u_{3}}^{n}]$ is $n {u_{3}}^{n - 1}$ where $n = 2$ .

$81 e^{9 x} - 14 \cos (x) \sin (x) - 7 (2 u_{3} \frac{d}{d x} [\sin (x)])$

Step 2.4.2.3

Replace all occurrences of $u_{3}$ with $\sin (x)$ .

$81 e^{9 x} - 14 \cos (x) \sin (x) - 7 (2 \sin (x) \frac{d}{d x} [\sin (x)])$

Step 2.4.3

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$81 e^{9 x} - 14 \cos (x) \sin (x) - 7 (2 \sin (x) \cos (x))$

Step 2.4.4

Multiply $2$ by $- 7$ .

$81 e^{9 x} - 14 \cos (x) \sin (x) - 14 \sin (x) \cos (x)$

Step 2.5

Simplify.

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Step 2.5.1

Combine terms.

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Step 2.5.1.1

Reorder the factors of $- 14 \sin (x) \cos (x)$ .

$81 e^{9 x} - 14 \cos (x) \sin (x) - 14 \cos (x) \sin (x)$

Step 2.5.1.2

Subtract $14 \cos (x) \sin (x)$ from $- 14 \cos (x) \sin (x)$ .

$81 e^{9 x} - 28 \cos (x) \sin (x)$

Step 2.5.2

Reorder terms.

$f'' (x) = - 28 \cos (x) \sin (x) + 81 e^{9 x}$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $- 28 \cos (x) \sin (x) + 81 e^{9 x}$ with respect to $x$ is $\frac{d}{d x} [- 28 \cos (x) \sin (x)] + \frac{d}{d x} [81 e^{9 x}]$ .

$\frac{d}{d x} [- 28 \cos (x) \sin (x)] + \frac{d}{d x} [81 e^{9 x}]$

Step 3.2

Evaluate $\frac{d}{d x} [- 28 \cos (x) \sin (x)]$ .

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Step 3.2.1

Since $- 28$ is constant with respect to $x$ , the derivative of $- 28 \cos (x) \sin (x)$ with respect to $x$ is $- 28 \frac{d}{d x} [\cos (x) \sin (x)]$ .

$- 28 \frac{d}{d x} [\cos (x) \sin (x)] + \frac{d}{d x} [81 e^{9 x}]$

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 $f (x) = \cos (x)$ and $g (x) = \sin (x)$ .

$- 28 (\cos (x) \frac{d}{d x} [\sin (x)] + \sin (x) \frac{d}{d x} [\cos (x)]) + \frac{d}{d x} [81 e^{9 x}]$

Step 3.2.3

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$- 28 (\cos (x) \cos (x) + \sin (x) \frac{d}{d x} [\cos (x)]) + \frac{d}{d x} [81 e^{9 x}]$

Step 3.2.4

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$- 28 (\cos (x) \cos (x) + \sin (x) (- \sin (x))) + \frac{d}{d x} [81 e^{9 x}]$

Step 3.2.5

Raise $\cos (x)$ to the power of $1$ .

$- 28 (\cos^{1} (x) \cos (x) + \sin (x) (- \sin (x))) + \frac{d}{d x} [81 e^{9 x}]$

Step 3.2.6

Raise $\cos (x)$ to the power of $1$ .

$- 28 (\cos^{1} (x) \cos^{1} (x) + \sin (x) (- \sin (x))) + \frac{d}{d x} [81 e^{9 x}]$

Step 3.2.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 28 ({\cos (x)}^{1 + 1} + \sin (x) (- \sin (x))) + \frac{d}{d x} [81 e^{9 x}]$

Step 3.2.8

Add $1$ and $1$ .

$- 28 (\cos^{2} (x) + \sin (x) (- \sin (x))) + \frac{d}{d x} [81 e^{9 x}]$

Step 3.2.9

Raise $\sin (x)$ to the power of $1$ .

$- 28 (\cos^{2} (x) - (\sin^{1} (x) \sin (x))) + \frac{d}{d x} [81 e^{9 x}]$

Step 3.2.10

Raise $\sin (x)$ to the power of $1$ .

$- 28 (\cos^{2} (x) - (\sin^{1} (x) \sin^{1} (x))) + \frac{d}{d x} [81 e^{9 x}]$

Step 3.2.11

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 28 (\cos^{2} (x) - {\sin (x)}^{1 + 1}) + \frac{d}{d x} [81 e^{9 x}]$

Step 3.2.12

Add $1$ and $1$ .

$- 28 (\cos^{2} (x) - \sin^{2} (x)) + \frac{d}{d x} [81 e^{9 x}]$

Step 3.3

Evaluate $\frac{d}{d x} [81 e^{9 x}]$ .

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Step 3.3.1

Since $81$ is constant with respect to $x$ , the derivative of $81 e^{9 x}$ with respect to $x$ is $81 \frac{d}{d x} [e^{9 x}]$ .

$- 28 (\cos^{2} (x) - \sin^{2} (x)) + 81 \frac{d}{d x} [e^{9 x}]$

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 $g (x) = 9 x$ .

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Step 3.3.2.1

To apply the Chain Rule, set $u$ as $9 x$ .

$- 28 (\cos^{2} (x) - \sin^{2} (x)) + 81 (\frac{d}{d u} [e^{u}] \frac{d}{d x} [9 x])$

Step 3.3.2.2

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

$- 28 (\cos^{2} (x) - \sin^{2} (x)) + 81 (e^{u} \frac{d}{d x} [9 x])$

Step 3.3.2.3

Replace all occurrences of $u$ with $9 x$ .

$- 28 (\cos^{2} (x) - \sin^{2} (x)) + 81 (e^{9 x} \frac{d}{d x} [9 x])$

Step 3.3.3

Since $9$ is constant with respect to $x$ , the derivative of $9 x$ with respect to $x$ is $9 \frac{d}{d x} [x]$ .

$- 28 (\cos^{2} (x) - \sin^{2} (x)) + 81 (e^{9 x} (9 \frac{d}{d x} [x]))$

Step 3.3.4

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

$- 28 (\cos^{2} (x) - \sin^{2} (x)) + 81 (e^{9 x} (9 \cdot 1))$

Step 3.3.5

Multiply $9$ by $1$ .

$- 28 (\cos^{2} (x) - \sin^{2} (x)) + 81 (e^{9 x} \cdot 9)$

Step 3.3.6

Move $9$ to the left of $e^{9 x}$ .

$- 28 (\cos^{2} (x) - \sin^{2} (x)) + 81 (9 \cdot e^{9 x})$

Step 3.3.7

Multiply $9$ by $81$ .

$- 28 (\cos^{2} (x) - \sin^{2} (x)) + 729 e^{9 x}$

Step 3.4

Simplify.

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Step 3.4.1

Apply the distributive property.

$- 28 \cos^{2} (x) - 28 (- \sin^{2} (x)) + 729 e^{9 x}$

Step 3.4.2

Multiply $- 1$ by $- 28$ .

$- 28 \cos^{2} (x) + 28 \sin^{2} (x) + 729 e^{9 x}$

Step 3.4.3

Reorder terms.

$f''' (x) = 729 e^{9 x} - 28 \cos^{2} (x) + 28 \sin^{2} (x)$

Step 4

Find the fourth derivative.

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Step 4.1

By the Sum Rule, the derivative of $729 e^{9 x} - 28 \cos^{2} (x) + 28 \sin^{2} (x)$ with respect to $x$ is $\frac{d}{d x} [729 e^{9 x}] + \frac{d}{d x} [- 28 \cos^{2} (x)] + \frac{d}{d x} [28 \sin^{2} (x)]$ .

$\frac{d}{d x} [729 e^{9 x}] + \frac{d}{d x} [- 28 \cos^{2} (x)] + \frac{d}{d x} [28 \sin^{2} (x)]$

Step 4.2

Evaluate $\frac{d}{d x} [729 e^{9 x}]$ .

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Step 4.2.1

Since $729$ is constant with respect to $x$ , the derivative of $729 e^{9 x}$ with respect to $x$ is $729 \frac{d}{d x} [e^{9 x}]$ .

$729 \frac{d}{d x} [e^{9 x}] + \frac{d}{d x} [- 28 \cos^{2} (x)] + \frac{d}{d x} [28 \sin^{2} (x)]$

Step 4.2.2

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 $g (x) = 9 x$ .

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Step 4.2.2.1

To apply the Chain Rule, set $u_{1}$ as $9 x$ .

$729 (\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [9 x]) + \frac{d}{d x} [- 28 \cos^{2} (x)] + \frac{d}{d x} [28 \sin^{2} (x)]$

Step 4.2.2.2

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

$729 (e^{u_{1}} \frac{d}{d x} [9 x]) + \frac{d}{d x} [- 28 \cos^{2} (x)] + \frac{d}{d x} [28 \sin^{2} (x)]$

Step 4.2.2.3

Replace all occurrences of $u_{1}$ with $9 x$ .

$729 (e^{9 x} \frac{d}{d x} [9 x]) + \frac{d}{d x} [- 28 \cos^{2} (x)] + \frac{d}{d x} [28 \sin^{2} (x)]$

Step 4.2.3

Since $9$ is constant with respect to $x$ , the derivative of $9 x$ with respect to $x$ is $9 \frac{d}{d x} [x]$ .

$729 (e^{9 x} (9 \frac{d}{d x} [x])) + \frac{d}{d x} [- 28 \cos^{2} (x)] + \frac{d}{d x} [28 \sin^{2} (x)]$

Step 4.2.4

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

$729 (e^{9 x} (9 \cdot 1)) + \frac{d}{d x} [- 28 \cos^{2} (x)] + \frac{d}{d x} [28 \sin^{2} (x)]$

Step 4.2.5

Multiply $9$ by $1$ .

$729 (e^{9 x} \cdot 9) + \frac{d}{d x} [- 28 \cos^{2} (x)] + \frac{d}{d x} [28 \sin^{2} (x)]$

Step 4.2.6

Move $9$ to the left of $e^{9 x}$ .

$729 (9 \cdot e^{9 x}) + \frac{d}{d x} [- 28 \cos^{2} (x)] + \frac{d}{d x} [28 \sin^{2} (x)]$

Step 4.2.7

Multiply $9$ by $729$ .

$6561 e^{9 x} + \frac{d}{d x} [- 28 \cos^{2} (x)] + \frac{d}{d x} [28 \sin^{2} (x)]$

Step 4.3

Evaluate $\frac{d}{d x} [- 28 \cos^{2} (x)]$ .

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Step 4.3.1

Since $- 28$ is constant with respect to $x$ , the derivative of $- 28 \cos^{2} (x)$ with respect to $x$ is $- 28 \frac{d}{d x} [\cos^{2} (x)]$ .

$6561 e^{9 x} - 28 \frac{d}{d x} [\cos^{2} (x)] + \frac{d}{d x} [28 \sin^{2} (x)]$

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

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Step 4.3.2.1

To apply the Chain Rule, set $u_{2}$ as $\cos (x)$ .

$6561 e^{9 x} - 28 (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [\cos (x)]) + \frac{d}{d x} [28 \sin^{2} (x)]$

Step 4.3.2.2

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

$6561 e^{9 x} - 28 (2 u_{2} \frac{d}{d x} [\cos (x)]) + \frac{d}{d x} [28 \sin^{2} (x)]$

Step 4.3.2.3

Replace all occurrences of $u_{2}$ with $\cos (x)$ .

$6561 e^{9 x} - 28 (2 \cos (x) \frac{d}{d x} [\cos (x)]) + \frac{d}{d x} [28 \sin^{2} (x)]$

Step 4.3.3

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$6561 e^{9 x} - 28 (2 \cos (x) (- \sin (x))) + \frac{d}{d x} [28 \sin^{2} (x)]$

Step 4.3.4

Multiply $- 1$ by $2$ .

$6561 e^{9 x} - 28 (- 2 \cos (x) \sin (x)) + \frac{d}{d x} [28 \sin^{2} (x)]$

Step 4.3.5

Multiply $- 2$ by $- 28$ .

$6561 e^{9 x} + 56 \cos (x) \sin (x) + \frac{d}{d x} [28 \sin^{2} (x)]$

Step 4.4

Evaluate $\frac{d}{d x} [28 \sin^{2} (x)]$ .

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Step 4.4.1

Since $28$ is constant with respect to $x$ , the derivative of $28 \sin^{2} (x)$ with respect to $x$ is $28 \frac{d}{d x} [\sin^{2} (x)]$ .

$6561 e^{9 x} + 56 \cos (x) \sin (x) + 28 \frac{d}{d x} [\sin^{2} (x)]$

Step 4.4.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \sin (x)$ .

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Step 4.4.2.1

To apply the Chain Rule, set $u_{3}$ as $\sin (x)$ .

$6561 e^{9 x} + 56 \cos (x) \sin (x) + 28 (\frac{d}{d u_{3}} [{u_{3}}^{2}] \frac{d}{d x} [\sin (x)])$

Step 4.4.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{3}} [{u_{3}}^{n}]$ is $n {u_{3}}^{n - 1}$ where $n = 2$ .

$6561 e^{9 x} + 56 \cos (x) \sin (x) + 28 (2 u_{3} \frac{d}{d x} [\sin (x)])$

Step 4.4.2.3

Replace all occurrences of $u_{3}$ with $\sin (x)$ .

$6561 e^{9 x} + 56 \cos (x) \sin (x) + 28 (2 \sin (x) \frac{d}{d x} [\sin (x)])$

Step 4.4.3

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$6561 e^{9 x} + 56 \cos (x) \sin (x) + 28 (2 \sin (x) \cos (x))$

Step 4.4.4

Multiply $2$ by $28$ .

$6561 e^{9 x} + 56 \cos (x) \sin (x) + 56 \sin (x) \cos (x)$

Step 4.5

Simplify.

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Step 4.5.1

Combine terms.

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Step 4.5.1.1

Reorder the factors of $56 \sin (x) \cos (x)$ .

$6561 e^{9 x} + 56 \cos (x) \sin (x) + 56 \cos (x) \sin (x)$

Step 4.5.1.2

Add $56 \cos (x) \sin (x)$ and $56 \cos (x) \sin (x)$ .

$6561 e^{9 x} + 112 \cos (x) \sin (x)$

Step 4.5.2

Reorder terms.

$f^{4} (x) = 112 \cos (x) \sin (x) + 6561 e^{9 x}$

Step 5

The fourth derivative of $f (x)$ with respect to $x$ is $112 \cos (x) \sin (x) + 6561 e^{9 x}$ .

$112 \cos (x) \sin (x) + 6561 e^{9 x}$