Calculus Examples

$y = \csc (x)$

Step 1

Find the first derivative.

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Step 1.1

The derivative of $\csc (x)$ with respect to $x$ is $- \csc (x) \cot (x)$ .

$f' (x) = - \csc (x) \cot (x)$

Step 1.2

Reorder the factors of $- \csc (x) \cot (x)$ .

$f' (x) = - \cot (x) \csc (x)$

Step 2

Find the second derivative.

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Step 2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \cot (x) \csc (x)$ with respect to $x$ is $- \frac{d}{d x} [\cot (x) \csc (x)]$ .

$f'' (x) = - \frac{d}{d x} (\cot (x) \csc (x))$

Step 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 $f (x) = \cot (x)$ and $g (x) = \csc (x)$ .

$f'' (x) = - (\cot (x) \frac{d}{d x} (\csc (x)) + \csc (x) \frac{d}{d x} (\cot (x)))$

Step 2.3

The derivative of $\csc (x)$ with respect to $x$ is $- \csc (x) \cot (x)$ .

$f'' (x) = - (\cot (x) (- \csc (x) \cot (x)) + \csc (x) \frac{d}{d x} (\cot (x)))$

Step 2.4

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

$f'' (x) = - (- \csc (x) (\cot (x) \cot (x)) + \csc (x) \frac{d}{d x} (\cot (x)))$

Step 2.5

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

$f'' (x) = - (- \csc (x) (\cot (x) \cot (x)) + \csc (x) \frac{d}{d x} (\cot (x)))$

Step 2.6

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

$f'' (x) = - (- \csc (x) {\cot (x)}^{1 + 1} + \csc (x) \frac{d}{d x} (\cot (x)))$

Step 2.7

Add $1$ and $1$ .

$f'' (x) = - (- \csc (x) \cot^{2} (x) + \csc (x) \frac{d}{d x} (\cot (x)))$

Step 2.8

The derivative of $\cot (x)$ with respect to $x$ is $- \csc^{2} (x)$ .

$f'' (x) = - (- \csc (x) \cot^{2} (x) + \csc (x) (- \csc^{2} (x)))$

Step 2.9

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

$f'' (x) = - (- \csc (x) \cot^{2} (x) - (\csc (x) \csc^{2} (x)))$

Step 2.10

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

$f'' (x) = - (- \csc (x) \cot^{2} (x) - {\csc (x)}^{1 + 2})$

Step 2.11

Add $1$ and $2$ .

$f'' (x) = - (- \csc (x) \cot^{2} (x) - \csc^{3} (x))$

Step 2.12

Simplify.

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Step 2.12.1

Apply the distributive property.

$f'' (x) = - (- \csc (x) \cot^{2} (x)) + \csc^{3} (x)$

Step 2.12.2

Combine terms.

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Step 2.12.2.1

Multiply $- 1$ by $- 1$ .

$f'' (x) = 1 (\csc (x) \cot^{2} (x)) + \csc^{3} (x)$

Step 2.12.2.2

Multiply $\csc (x)$ by $1$ .

$f'' (x) = \csc (x) \cot^{2} (x) + \csc^{3} (x)$

Step 2.12.2.3

Multiply $- 1$ by $- 1$ .

$f'' (x) = \csc (x) \cot^{2} (x) + 1 \csc^{3} (x)$

Step 2.12.2.4

Multiply $\csc^{3} (x)$ by $1$ .

$f'' (x) = \csc (x) \cot^{2} (x) + \csc^{3} (x)$

Step 2.12.3

Reorder terms.

$f'' (x) = \cot^{2} (x) \csc (x) + \csc^{3} (x)$

Step 3

Find the third derivative.

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Step 3.1

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

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

Step 3.2

Evaluate $\frac{d}{d x} [\cot^{2} (x) \csc (x)]$ .

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

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

Step 3.2.2

The derivative of $\csc (x)$ with respect to $x$ is $- \csc (x) \cot (x)$ .

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

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

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Step 3.2.3.1

To apply the Chain Rule, set $u_{1}$ as $\cot (x)$ .

$f''' (x) = \cot^{2} (x) (- \csc (x) \cot (x)) + \csc (x) (\frac{d}{d u (1)} ({u_{1}}^{2}) \frac{d}{d x} (\cot (x))) + \frac{d}{d x} (\csc^{3} (x))$

Step 3.2.3.2

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

$f''' (x) = \cot^{2} (x) (- \csc (x) \cot (x)) + \csc (x) (2 u_{1} \frac{d}{d x} (\cot (x))) + \frac{d}{d x} (\csc^{3} (x))$

Step 3.2.3.3

Replace all occurrences of $u_{1}$ with $\cot (x)$ .

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

Step 3.2.4

The derivative of $\cot (x)$ with respect to $x$ is $- \csc^{2} (x)$ .

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

Step 3.2.5

Multiply $\cot^{2} (x)$ by $\cot (x)$ by adding the exponents.

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Step 3.2.5.1

Move $\cot (x)$ .

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

Step 3.2.5.2

Multiply $\cot (x)$ by $\cot^{2} (x)$ .

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Step 3.2.5.2.1

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

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

Step 3.2.5.2.2

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

$f''' (x) = {\cot (x)}^{1 + 2} (- \csc (x)) + \csc (x) (2 \cot (x) (- \csc^{2} (x))) + \frac{d}{d x} (\csc^{3} (x))$

Step 3.2.5.3

Add $1$ and $2$ .

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

Step 3.2.6

Multiply $- 1$ by $2$ .

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

Step 3.2.7

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

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

Step 3.2.8

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

$f''' (x) = \cot^{3} (x) (- \csc (x)) - 2 \cot (x) {\csc (x)}^{1 + 2} + \frac{d}{d x} (\csc^{3} (x))$

Step 3.2.9

Add $1$ and $2$ .

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

Step 3.3

Evaluate $\frac{d}{d x} [\csc^{3} (x)]$ .

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Step 3.3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = \csc (x)$ .

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Step 3.3.1.1

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

$f''' (x) = \cot^{3} (x) (- \csc (x)) - 2 \cot (x) \csc^{3} (x) + \frac{d}{d u (2)} ({u_{2}}^{3}) \frac{d}{d x} (\csc (x))$

Step 3.3.1.2

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

$f''' (x) = \cot^{3} (x) (- \csc (x)) - 2 \cot (x) \csc^{3} (x) + 3 {u_{2}}^{2} \frac{d}{d x} (\csc (x))$

Step 3.3.1.3

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

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

Step 3.3.2

The derivative of $\csc (x)$ with respect to $x$ is $- \csc (x) \cot (x)$ .

$f''' (x) = \cot^{3} (x) (- \csc (x)) - 2 \cot (x) \csc^{3} (x) + 3 \csc^{2} (x) (- \csc (x) \cot (x))$

Step 3.3.3

Multiply $- 1$ by $3$ .

$f''' (x) = \cot^{3} (x) (- \csc (x)) - 2 \cot (x) \csc^{3} (x) - 3 \csc^{2} (x) (\csc (x) \cot (x))$

Step 3.3.4

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

$f''' (x) = \cot^{3} (x) (- \csc (x)) - 2 \cot (x) \csc^{3} (x) - 3 (\csc (x) \csc^{2} (x)) \cot (x)$

Step 3.3.5

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

$f''' (x) = \cot^{3} (x) (- \csc (x)) - 2 \cot (x) \csc^{3} (x) - 3 {\csc (x)}^{1 + 2} \cot (x)$

Step 3.3.6

Add $1$ and $2$ .

$f''' (x) = \cot^{3} (x) (- \csc (x)) - 2 \cot (x) \csc^{3} (x) - 3 \csc^{3} (x) \cot (x)$

Step 3.4

Simplify.

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Step 3.4.1

Combine terms.

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Step 3.4.1.1

Reorder the factors of $- 2 \cot (x) \csc^{3} (x)$ .

$f''' (x) = \cot^{3} (x) (- \csc (x)) - 2 \csc^{3} (x) \cot (x) - 3 \csc^{3} (x) \cot (x)$

Step 3.4.1.2

Subtract $3 \csc^{3} (x) \cot (x)$ from $- 2 \csc^{3} (x) \cot (x)$ .

$f''' (x) = \cot^{3} (x) (- \csc (x)) - 5 \csc^{3} (x) \cot (x)$

Step 3.4.2

Reorder terms.

$f''' (x) = - \cot^{3} (x) \csc (x) - 5 \csc^{3} (x) \cot (x)$

Step 4

Find the fourth derivative.

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Step 4.1

By the Sum Rule, the derivative of $- \cot^{3} (x) \csc (x) - 5 \csc^{3} (x) \cot (x)$ with respect to $x$ is $\frac{d}{d x} [- \cot^{3} (x) \csc (x)] + \frac{d}{d x} [- 5 \csc^{3} (x) \cot (x)]$ .

$f^{4} (x) = \frac{d}{d x} (- \cot^{3} (x) \csc (x)) + \frac{d}{d x} (- 5 \csc^{3} (x) \cot (x))$

Step 4.2

Evaluate $\frac{d}{d x} [- \cot^{3} (x) \csc (x)]$ .

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Step 4.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \cot^{3} (x) \csc (x)$ with respect to $x$ is $- \frac{d}{d x} [\cot^{3} (x) \csc (x)]$ .

$f^{4} (x) = - \frac{d}{d x} (\cot^{3} (x) \csc (x)) + \frac{d}{d x} (- 5 \csc^{3} (x) \cot (x))$

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 $f (x) = \cot^{3} (x)$ and $g (x) = \csc (x)$ .

$f^{4} (x) = - (\cot^{3} (x) \frac{d}{d x} (\csc (x)) + \csc (x) \frac{d}{d x} (\cot^{3} (x))) + \frac{d}{d x} (- 5 \csc^{3} (x) \cot (x))$

Step 4.2.3

The derivative of $\csc (x)$ with respect to $x$ is $- \csc (x) \cot (x)$ .

$f^{4} (x) = - (\cot^{3} (x) (- \csc (x) \cot (x)) + \csc (x) \frac{d}{d x} (\cot^{3} (x))) + \frac{d}{d x} (- 5 \csc^{3} (x) \cot (x))$

Step 4.2.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = \cot (x)$ .

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Step 4.2.4.1

To apply the Chain Rule, set $u_{1}$ as $\cot (x)$ .

$f^{4} (x) = - (\cot^{3} (x) (- \csc (x) \cot (x)) + \csc (x) (\frac{d}{d u (1)} ({u_{1}}^{3}) \frac{d}{d x} (\cot (x)))) + \frac{d}{d x} (- 5 \csc^{3} (x) \cot (x))$

Step 4.2.4.2

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

$f^{4} (x) = - (\cot^{3} (x) (- \csc (x) \cot (x)) + \csc (x) (3 {u_{1}}^{2} \frac{d}{d x} (\cot (x)))) + \frac{d}{d x} (- 5 \csc^{3} (x) \cot (x))$

Step 4.2.4.3

Replace all occurrences of $u_{1}$ with $\cot (x)$ .

$f^{4} (x) = - (\cot^{3} (x) (- \csc (x) \cot (x)) + \csc (x) (3 \cot^{2} (x) \frac{d}{d x} (\cot (x)))) + \frac{d}{d x} (- 5 \csc^{3} (x) \cot (x))$

Step 4.2.5

The derivative of $\cot (x)$ with respect to $x$ is $- \csc^{2} (x)$ .

$f^{4} (x) = - (\cot^{3} (x) (- \csc (x) \cot (x)) + \csc (x) (3 \cot^{2} (x) (- \csc^{2} (x)))) + \frac{d}{d x} (- 5 \csc^{3} (x) \cot (x))$

Step 4.2.6

Multiply $\cot^{3} (x)$ by $\cot (x)$ by adding the exponents.

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Step 4.2.6.1

Move $\cot (x)$ .

$f^{4} (x) = - (\cot (x) \cot^{3} (x) (- \csc (x)) + \csc (x) (3 \cot^{2} (x) (- \csc^{2} (x)))) + \frac{d}{d x} (- 5 \csc^{3} (x) \cot (x))$

Step 4.2.6.2

Multiply $\cot (x)$ by $\cot^{3} (x)$ .

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Step 4.2.6.2.1

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

$f^{4} (x) = - (\cot (x) \cot^{3} (x) (- \csc (x)) + \csc (x) (3 \cot^{2} (x) (- \csc^{2} (x)))) + \frac{d}{d x} (- 5 \csc^{3} (x) \cot (x))$

Step 4.2.6.2.2

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

$f^{4} (x) = - ({\cot (x)}^{1 + 3} (- \csc (x)) + \csc (x) (3 \cot^{2} (x) (- \csc^{2} (x)))) + \frac{d}{d x} (- 5 \csc^{3} (x) \cot (x))$

Step 4.2.6.3

Add $1$ and $3$ .

$f^{4} (x) = - (\cot^{4} (x) (- \csc (x)) + \csc (x) (3 \cot^{2} (x) (- \csc^{2} (x)))) + \frac{d}{d x} (- 5 \csc^{3} (x) \cot (x))$

Step 4.2.7

Multiply $- 1$ by $3$ .

$f^{4} (x) = - (\cot^{4} (x) (- \csc (x)) + \csc (x) (- 3 \cot^{2} (x) \csc^{2} (x))) + \frac{d}{d x} (- 5 \csc^{3} (x) \cot (x))$

Step 4.2.8

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

$f^{4} (x) = - (\cot^{4} (x) (- \csc (x)) - 3 \cot^{2} (x) (\csc (x) \csc^{2} (x))) + \frac{d}{d x} (- 5 \csc^{3} (x) \cot (x))$

Step 4.2.9

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

$f^{4} (x) = - (\cot^{4} (x) (- \csc (x)) - 3 \cot^{2} (x) {\csc (x)}^{1 + 2}) + \frac{d}{d x} (- 5 \csc^{3} (x) \cot (x))$

Step 4.2.10

Add $1$ and $2$ .

$f^{4} (x) = - (\cot^{4} (x) (- \csc (x)) - 3 \cot^{2} (x) \csc^{3} (x)) + \frac{d}{d x} (- 5 \csc^{3} (x) \cot (x))$

Step 4.3

Evaluate $\frac{d}{d x} [- 5 \csc^{3} (x) \cot (x)]$ .

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Step 4.3.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 \csc^{3} (x) \cot (x)$ with respect to $x$ is $- 5 \frac{d}{d x} [\csc^{3} (x) \cot (x)]$ .

$f^{4} (x) = - (\cot^{4} (x) (- \csc (x)) - 3 \cot^{2} (x) \csc^{3} (x)) - 5 \frac{d}{d x} (\csc^{3} (x) \cot (x))$

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 $f (x) = \csc^{3} (x)$ and $g (x) = \cot (x)$ .

$f^{4} (x) = - (\cot^{4} (x) (- \csc (x)) - 3 \cot^{2} (x) \csc^{3} (x)) - 5 (\csc^{3} (x) \frac{d}{d x} (\cot (x)) + \cot (x) \frac{d}{d x} (\csc^{3} (x)))$

Step 4.3.3

The derivative of $\cot (x)$ with respect to $x$ is $- \csc^{2} (x)$ .

$f^{4} (x) = - (\cot^{4} (x) (- \csc (x)) - 3 \cot^{2} (x) \csc^{3} (x)) - 5 (\csc^{3} (x) (- \csc^{2} (x)) + \cot (x) \frac{d}{d x} (\csc^{3} (x)))$

Step 4.3.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = \csc (x)$ .

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Step 4.3.4.1

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

$f^{4} (x) = - (\cot^{4} (x) (- \csc (x)) - 3 \cot^{2} (x) \csc^{3} (x)) - 5 (\csc^{3} (x) (- \csc^{2} (x)) + \cot (x) (\frac{d}{d u (2)} ({u_{2}}^{3}) \frac{d}{d x} (\csc (x))))$

Step 4.3.4.2

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

$f^{4} (x) = - (\cot^{4} (x) (- \csc (x)) - 3 \cot^{2} (x) \csc^{3} (x)) - 5 (\csc^{3} (x) (- \csc^{2} (x)) + \cot (x) (3 {u_{2}}^{2} \frac{d}{d x} (\csc (x))))$

Step 4.3.4.3

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

$f^{4} (x) = - (\cot^{4} (x) (- \csc (x)) - 3 \cot^{2} (x) \csc^{3} (x)) - 5 (\csc^{3} (x) (- \csc^{2} (x)) + \cot (x) (3 \csc^{2} (x) \frac{d}{d x} (\csc (x))))$

Step 4.3.5

The derivative of $\csc (x)$ with respect to $x$ is $- \csc (x) \cot (x)$ .

$f^{4} (x) = - (\cot^{4} (x) (- \csc (x)) - 3 \cot^{2} (x) \csc^{3} (x)) - 5 (\csc^{3} (x) (- \csc^{2} (x)) + \cot (x) (3 \csc^{2} (x) (- \csc (x) \cot (x))))$

Step 4.3.6

Multiply $\csc^{3} (x)$ by $\csc^{2} (x)$ by adding the exponents.

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Step 4.3.6.1

Move $\csc^{2} (x)$ .

$f^{4} (x) = - (\cot^{4} (x) (- \csc (x)) - 3 \cot^{2} (x) \csc^{3} (x)) - 5 (\csc^{2} (x) \csc^{3} (x) \cdot - 1 + \cot (x) (3 \csc^{2} (x) (- \csc (x) \cot (x))))$

Step 4.3.6.2

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

$f^{4} (x) = - (\cot^{4} (x) (- \csc (x)) - 3 \cot^{2} (x) \csc^{3} (x)) - 5 ({\csc (x)}^{2 + 3} \cdot - 1 + \cot (x) (3 \csc^{2} (x) (- \csc (x) \cot (x))))$

Step 4.3.6.3

Add $2$ and $3$ .

$f^{4} (x) = - (\cot^{4} (x) (- \csc (x)) - 3 \cot^{2} (x) \csc^{3} (x)) - 5 (\csc^{5} (x) \cdot - 1 + \cot (x) (3 \csc^{2} (x) (- \csc (x) \cot (x))))$

Step 4.3.7

Move $- 1$ to the left of $\csc^{5} (x)$ .

$f^{4} (x) = - (\cot^{4} (x) (- \csc (x)) - 3 \cot^{2} (x) \csc^{3} (x)) - 5 (- 1 \cdot \csc^{5} (x) + \cot (x) (3 \csc^{2} (x) (- \csc (x) \cot (x))))$

Step 4.3.8

Rewrite $- 1 \csc^{5} (x)$ as $- \csc^{5} (x)$ .

$f^{4} (x) = - (\cot^{4} (x) (- \csc (x)) - 3 \cot^{2} (x) \csc^{3} (x)) - 5 (- \csc^{5} (x) + \cot (x) (3 \csc^{2} (x) (- \csc (x) \cot (x))))$

Step 4.3.9

Multiply $- 1$ by $3$ .

$f^{4} (x) = - (\cot^{4} (x) (- \csc (x)) - 3 \cot^{2} (x) \csc^{3} (x)) - 5 (- \csc^{5} (x) + \cot (x) (- 3 \csc^{2} (x) (\csc (x) \cot (x))))$

Step 4.3.10

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

$f^{4} (x) = - (\cot^{4} (x) (- \csc (x)) - 3 \cot^{2} (x) \csc^{3} (x)) - 5 (- \csc^{5} (x) + \cot (x) (- 3 (\csc (x) \csc^{2} (x)) \cot (x)))$

Step 4.3.11

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

$f^{4} (x) = - (\cot^{4} (x) (- \csc (x)) - 3 \cot^{2} (x) \csc^{3} (x)) - 5 (- \csc^{5} (x) + \cot (x) (- 3 {\csc (x)}^{1 + 2} \cot (x)))$

Step 4.3.12

Add $1$ and $2$ .

$f^{4} (x) = - (\cot^{4} (x) (- \csc (x)) - 3 \cot^{2} (x) \csc^{3} (x)) - 5 (- \csc^{5} (x) + \cot (x) (- 3 \csc^{3} (x) \cot (x)))$

Step 4.3.13

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

$f^{4} (x) = - (\cot^{4} (x) (- \csc (x)) - 3 \cot^{2} (x) \csc^{3} (x)) - 5 (- \csc^{5} (x) - 3 \csc^{3} (x) (\cot (x) \cot (x)))$

Step 4.3.14

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

$f^{4} (x) = - (\cot^{4} (x) (- \csc (x)) - 3 \cot^{2} (x) \csc^{3} (x)) - 5 (- \csc^{5} (x) - 3 \csc^{3} (x) (\cot (x) \cot (x)))$

Step 4.3.15

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

$f^{4} (x) = - (\cot^{4} (x) (- \csc (x)) - 3 \cot^{2} (x) \csc^{3} (x)) - 5 (- \csc^{5} (x) - 3 \csc^{3} (x) {\cot (x)}^{1 + 1})$

Step 4.3.16

Add $1$ and $1$ .

$f^{4} (x) = - (\cot^{4} (x) (- \csc (x)) - 3 \cot^{2} (x) \csc^{3} (x)) - 5 (- \csc^{5} (x) - 3 \csc^{3} (x) \cot^{2} (x))$

Step 4.4

Simplify.

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Step 4.4.1

Apply the distributive property.

$f^{4} (x) = - (\cot^{4} (x) (- \csc (x))) - (- 3 \cot^{2} (x) \csc^{3} (x)) - 5 (- \csc^{5} (x) - 3 \csc^{3} (x) \cot^{2} (x))$

Step 4.4.2

Apply the distributive property.

$f^{4} (x) = - (\cot^{4} (x) (- \csc (x))) - (- 3 \cot^{2} (x) \csc^{3} (x)) - 5 (- \csc^{5} (x)) - 5 (- 3 \csc^{3} (x) \cot^{2} (x))$

Step 4.4.3

Combine terms.

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Step 4.4.3.1

Multiply $- 1$ by $- 1$ .

$f^{4} (x) = 1 (\cot^{4} (x) (\csc (x))) - (- 3 \cot^{2} (x) \csc^{3} (x)) - 5 (- \csc^{5} (x)) - 5 (- 3 \csc^{3} (x) \cot^{2} (x))$

Step 4.4.3.2

Multiply $\cot^{4} (x)$ by $1$ .

$f^{4} (x) = \cot^{4} (x) \csc (x) - (- 3 \cot^{2} (x) \csc^{3} (x)) - 5 (- \csc^{5} (x)) - 5 (- 3 \csc^{3} (x) \cot^{2} (x))$

Step 4.4.3.3

Multiply $- 3$ by $- 1$ .

$f^{4} (x) = \cot^{4} (x) \csc (x) + 3 (\cot^{2} (x) \csc^{3} (x)) - 5 (- \csc^{5} (x)) - 5 (- 3 \csc^{3} (x) \cot^{2} (x))$

Step 4.4.3.4

Multiply $- 1$ by $- 5$ .

$f^{4} (x) = \cot^{4} (x) \csc (x) + 3 \cot^{2} (x) \csc^{3} (x) + 5 \csc^{5} (x) - 5 (- 3 \csc^{3} (x) \cot^{2} (x))$

Step 4.4.3.5

Multiply $- 3$ by $- 5$ .

$f^{4} (x) = \cot^{4} (x) \csc (x) + 3 \cot^{2} (x) \csc^{3} (x) + 5 \csc^{5} (x) + 15 (\csc^{3} (x) \cot^{2} (x))$

Step 4.4.3.6

Reorder the factors of $3 \cot^{2} (x) \csc^{3} (x)$ .

$f^{4} (x) = \cot^{4} (x) \csc (x) + 3 \csc^{3} (x) \cot^{2} (x) + 5 \csc^{5} (x) + 15 \csc^{3} (x) \cot^{2} (x)$

Step 4.4.3.7

Add $3 \csc^{3} (x) \cot^{2} (x)$ and $15 \csc^{3} (x) \cot^{2} (x)$ .

$f^{4} (x) = \cot^{4} (x) \csc (x) + 18 \csc^{3} (x) \cot^{2} (x) + 5 \csc^{5} (x)$