Calculus Examples

$\cot (θ)$

Step 1

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

$f' (θ) = - \csc^{2} (θ)$

Step 2

Find the second derivative.

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Step 2.1

Since $- 1$ is constant with respect to $θ$ , the derivative of $- \csc^{2} (θ)$ with respect to $θ$ is $- \frac{d}{d θ} [\csc^{2} (θ)]$ .

$- \frac{d}{d θ} [\csc^{2} (θ)]$

Step 2.2

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

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Step 2.2.1

To apply the Chain Rule, set $u$ as $\csc (θ)$ .

$- (\frac{d}{d u} [u^{2}] \frac{d}{d θ} [\csc (θ)])$

Step 2.2.2

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

$- (2 u \frac{d}{d θ} [\csc (θ)])$

Step 2.2.3

Replace all occurrences of $u$ with $\csc (θ)$ .

$- (2 \csc (θ) \frac{d}{d θ} [\csc (θ)])$

Step 2.3

Multiply $2$ by $- 1$ .

$- 2 (\csc (θ) \frac{d}{d θ} [\csc (θ)])$

Step 2.4

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

$- 2 \csc (θ) (- \csc (θ) \cot (θ))$

Step 2.5

Multiply $- 1$ by $- 2$ .

$2 \csc (θ) (\csc (θ) \cot (θ))$

Step 2.6

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

$2 (\csc^{1} (θ) \csc (θ)) \cot (θ)$

Step 2.7

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

$2 (\csc^{1} (θ) \csc^{1} (θ)) \cot (θ)$

Step 2.8

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

$2 {\csc (θ)}^{1 + 1} \cot (θ)$

Step 2.9

Add $1$ and $1$ .

$f'' (θ) = 2 \csc^{2} (θ) \cot (θ)$

Step 3

Find the third derivative.

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Step 3.1

Since $2$ is constant with respect to $θ$ , the derivative of $2 \csc^{2} (θ) \cot (θ)$ with respect to $θ$ is $2 \frac{d}{d θ} [\csc^{2} (θ) \cot (θ)]$ .

$2 \frac{d}{d θ} [\csc^{2} (θ) \cot (θ)]$

Step 3.2

Differentiate using the Product Rule which states that $\frac{d}{d θ} [f (θ) g (θ)]$ is $f (θ) \frac{d}{d θ} [g (θ)] + g (θ) \frac{d}{d θ} [f (θ)]$ where $f (θ) = \csc^{2} (θ)$ and $g (θ) = \cot (θ)$ .

$2 (\csc^{2} (θ) \frac{d}{d θ} [\cot (θ)] + \cot (θ) \frac{d}{d θ} [\csc^{2} (θ)])$

Step 3.3

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

$2 (\csc^{2} (θ) (- \csc^{2} (θ)) + \cot (θ) \frac{d}{d θ} [\csc^{2} (θ)])$

Step 3.4

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

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Step 3.4.1

Move $\csc^{2} (θ)$ .

$2 (\csc^{2} (θ) \csc^{2} (θ) \cdot - 1 + \cot (θ) \frac{d}{d θ} [\csc^{2} (θ)])$

Step 3.4.2

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

$2 ({\csc (θ)}^{2 + 2} \cdot - 1 + \cot (θ) \frac{d}{d θ} [\csc^{2} (θ)])$

Step 3.4.3

Add $2$ and $2$ .

$2 (\csc^{4} (θ) \cdot - 1 + \cot (θ) \frac{d}{d θ} [\csc^{2} (θ)])$

Step 3.5

Simplify the expression.

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Step 3.5.1

Move $- 1$ to the left of $\csc^{4} (θ)$ .

$2 (- 1 \cdot \csc^{4} (θ) + \cot (θ) \frac{d}{d θ} [\csc^{2} (θ)])$

Step 3.5.2

Rewrite $- 1 \csc^{4} (θ)$ as $- \csc^{4} (θ)$ .

$2 (- \csc^{4} (θ) + \cot (θ) \frac{d}{d θ} [\csc^{2} (θ)])$

Step 3.6

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

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Step 3.6.1

To apply the Chain Rule, set $u$ as $\csc (θ)$ .

$2 (- \csc^{4} (θ) + \cot (θ) (\frac{d}{d u} [u^{2}] \frac{d}{d θ} [\csc (θ)]))$

Step 3.6.2

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

$2 (- \csc^{4} (θ) + \cot (θ) (2 u \frac{d}{d θ} [\csc (θ)]))$

Step 3.6.3

Replace all occurrences of $u$ with $\csc (θ)$ .

$2 (- \csc^{4} (θ) + \cot (θ) (2 \csc (θ) \frac{d}{d θ} [\csc (θ)]))$

Step 3.7

Move $2$ to the left of $\cot (θ)$ .

$2 (- \csc^{4} (θ) + 2 \cdot \cot (θ) \csc (θ) \frac{d}{d θ} [\csc (θ)])$

Step 3.8

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

$2 (- \csc^{4} (θ) + 2 \cot (θ) \csc (θ) (- \csc (θ) \cot (θ)))$

Step 3.9

Multiply $- 1$ by $2$ .

$2 (- \csc^{4} (θ) - 2 \cot (θ) \csc (θ) (\csc (θ) \cot (θ)))$

Step 3.10

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

$2 (- \csc^{4} (θ) - 2 (\cot^{1} (θ) \cot (θ)) \csc (θ) \csc (θ))$

Step 3.11

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

$2 (- \csc^{4} (θ) - 2 (\cot^{1} (θ) \cot^{1} (θ)) \csc (θ) \csc (θ))$

Step 3.12

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

$2 (- \csc^{4} (θ) - 2 {\cot (θ)}^{1 + 1} \csc (θ) \csc (θ))$

Step 3.13

Add $1$ and $1$ .

$2 (- \csc^{4} (θ) - 2 \cot^{2} (θ) \csc (θ) \csc (θ))$

Step 3.14

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

$2 (- \csc^{4} (θ) - 2 \cot^{2} (θ) (\csc^{1} (θ) \csc (θ)))$

Step 3.15

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

$2 (- \csc^{4} (θ) - 2 \cot^{2} (θ) (\csc^{1} (θ) \csc^{1} (θ)))$

Step 3.16

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

$2 (- \csc^{4} (θ) - 2 \cot^{2} (θ) {\csc (θ)}^{1 + 1})$

Step 3.17

Add $1$ and $1$ .

$2 (- \csc^{4} (θ) - 2 \cot^{2} (θ) \csc^{2} (θ))$

Step 3.18

Simplify.

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Step 3.18.1

Apply the distributive property.

$2 (- \csc^{4} (θ)) + 2 (- 2 \cot^{2} (θ) \csc^{2} (θ))$

Step 3.18.2

Combine terms.

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Step 3.18.2.1

Multiply $- 1$ by $2$ .

$- 2 \csc^{4} (θ) + 2 (- 2 \cot^{2} (θ) \csc^{2} (θ))$

Step 3.18.2.2

Multiply $- 2$ by $2$ .

$- 2 \csc^{4} (θ) - 4 \cot^{2} (θ) \csc^{2} (θ)$

Step 3.18.3

Reorder terms.

$f''' (θ) = - 4 \cot^{2} (θ) \csc^{2} (θ) - 2 \csc^{4} (θ)$

Step 4

Find the fourth derivative.

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Step 4.1

By the Sum Rule, the derivative of $- 4 \cot^{2} (θ) \csc^{2} (θ) - 2 \csc^{4} (θ)$ with respect to $θ$ is $\frac{d}{d θ} [- 4 \cot^{2} (θ) \csc^{2} (θ)] + \frac{d}{d θ} [- 2 \csc^{4} (θ)]$ .

$\frac{d}{d θ} [- 4 \cot^{2} (θ) \csc^{2} (θ)] + \frac{d}{d θ} [- 2 \csc^{4} (θ)]$

Step 4.2

Evaluate $\frac{d}{d θ} [- 4 \cot^{2} (θ) \csc^{2} (θ)]$ .

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Step 4.2.1

Since $- 4$ is constant with respect to $θ$ , the derivative of $- 4 \cot^{2} (θ) \csc^{2} (θ)$ with respect to $θ$ is $- 4 \frac{d}{d θ} [\cot^{2} (θ) \csc^{2} (θ)]$ .

$- 4 \frac{d}{d θ} [\cot^{2} (θ) \csc^{2} (θ)] + \frac{d}{d θ} [- 2 \csc^{4} (θ)]$

Step 4.2.2

Differentiate using the Product Rule which states that $\frac{d}{d θ} [f (θ) g (θ)]$ is $f (θ) \frac{d}{d θ} [g (θ)] + g (θ) \frac{d}{d θ} [f (θ)]$ where $f (θ) = \cot^{2} (θ)$ and $g (θ) = \csc^{2} (θ)$ .

$- 4 (\cot^{2} (θ) \frac{d}{d θ} [\csc^{2} (θ)] + \csc^{2} (θ) \frac{d}{d θ} [\cot^{2} (θ)]) + \frac{d}{d θ} [- 2 \csc^{4} (θ)]$

Step 4.2.3

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

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Step 4.2.3.1

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

$- 4 (\cot^{2} (θ) (\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d θ} [\csc (θ)]) + \csc^{2} (θ) \frac{d}{d θ} [\cot^{2} (θ)]) + \frac{d}{d θ} [- 2 \csc^{4} (θ)]$

Step 4.2.3.2

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

$- 4 (\cot^{2} (θ) (2 u_{1} \frac{d}{d θ} [\csc (θ)]) + \csc^{2} (θ) \frac{d}{d θ} [\cot^{2} (θ)]) + \frac{d}{d θ} [- 2 \csc^{4} (θ)]$

Step 4.2.3.3

Replace all occurrences of $u_{1}$ with $\csc (θ)$ .

$- 4 (\cot^{2} (θ) (2 \csc (θ) \frac{d}{d θ} [\csc (θ)]) + \csc^{2} (θ) \frac{d}{d θ} [\cot^{2} (θ)]) + \frac{d}{d θ} [- 2 \csc^{4} (θ)]$

Step 4.2.4

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

$- 4 (\cot^{2} (θ) (2 \csc (θ) (- \csc (θ) \cot (θ))) + \csc^{2} (θ) \frac{d}{d θ} [\cot^{2} (θ)]) + \frac{d}{d θ} [- 2 \csc^{4} (θ)]$

Step 4.2.5

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

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Step 4.2.5.1

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

$- 4 (\cot^{2} (θ) (2 \csc (θ) (- \csc (θ) \cot (θ))) + \csc^{2} (θ) (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d θ} [\cot (θ)])) + \frac{d}{d θ} [- 2 \csc^{4} (θ)]$

Step 4.2.5.2

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

$- 4 (\cot^{2} (θ) (2 \csc (θ) (- \csc (θ) \cot (θ))) + \csc^{2} (θ) (2 u_{2} \frac{d}{d θ} [\cot (θ)])) + \frac{d}{d θ} [- 2 \csc^{4} (θ)]$

Step 4.2.5.3

Replace all occurrences of $u_{2}$ with $\cot (θ)$ .

$- 4 (\cot^{2} (θ) (2 \csc (θ) (- \csc (θ) \cot (θ))) + \csc^{2} (θ) (2 \cot (θ) \frac{d}{d θ} [\cot (θ)])) + \frac{d}{d θ} [- 2 \csc^{4} (θ)]$

Step 4.2.6

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

$- 4 (\cot^{2} (θ) (2 \csc (θ) (- \csc (θ) \cot (θ))) + \csc^{2} (θ) (2 \cot (θ) (- \csc^{2} (θ)))) + \frac{d}{d θ} [- 2 \csc^{4} (θ)]$

Step 4.2.7

Multiply $- 1$ by $2$ .

$- 4 (\cot^{2} (θ) (- 2 \csc (θ) (\csc (θ) \cot (θ))) + \csc^{2} (θ) (2 \cot (θ) (- \csc^{2} (θ)))) + \frac{d}{d θ} [- 2 \csc^{4} (θ)]$

Step 4.2.8

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

$- 4 (\cot^{2} (θ) (- 2 (\csc^{1} (θ) \csc (θ)) \cot (θ)) + \csc^{2} (θ) (2 \cot (θ) (- \csc^{2} (θ)))) + \frac{d}{d θ} [- 2 \csc^{4} (θ)]$

Step 4.2.9

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

$- 4 (\cot^{2} (θ) (- 2 (\csc^{1} (θ) \csc^{1} (θ)) \cot (θ)) + \csc^{2} (θ) (2 \cot (θ) (- \csc^{2} (θ)))) + \frac{d}{d θ} [- 2 \csc^{4} (θ)]$

Step 4.2.10

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

$- 4 (\cot^{2} (θ) (- 2 {\csc (θ)}^{1 + 1} \cot (θ)) + \csc^{2} (θ) (2 \cot (θ) (- \csc^{2} (θ)))) + \frac{d}{d θ} [- 2 \csc^{4} (θ)]$

Step 4.2.11

Add $1$ and $1$ .

$- 4 (\cot^{2} (θ) (- 2 \csc^{2} (θ) \cot (θ)) + \csc^{2} (θ) (2 \cot (θ) (- \csc^{2} (θ)))) + \frac{d}{d θ} [- 2 \csc^{4} (θ)]$

Step 4.2.12

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

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Step 4.2.12.1

Move $\cot (θ)$ .

$- 4 (\cot (θ) \cot^{2} (θ) (- 2 \csc^{2} (θ)) + \csc^{2} (θ) (2 \cot (θ) (- \csc^{2} (θ)))) + \frac{d}{d θ} [- 2 \csc^{4} (θ)]$

Step 4.2.12.2

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

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Step 4.2.12.2.1

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

$- 4 (\cot^{1} (θ) \cot^{2} (θ) (- 2 \csc^{2} (θ)) + \csc^{2} (θ) (2 \cot (θ) (- \csc^{2} (θ)))) + \frac{d}{d θ} [- 2 \csc^{4} (θ)]$

Step 4.2.12.2.2

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

$- 4 ({\cot (θ)}^{1 + 2} (- 2 \csc^{2} (θ)) + \csc^{2} (θ) (2 \cot (θ) (- \csc^{2} (θ)))) + \frac{d}{d θ} [- 2 \csc^{4} (θ)]$

Step 4.2.12.3

Add $1$ and $2$ .

$- 4 (\cot^{3} (θ) (- 2 \csc^{2} (θ)) + \csc^{2} (θ) (2 \cot (θ) (- \csc^{2} (θ)))) + \frac{d}{d θ} [- 2 \csc^{4} (θ)]$

Step 4.2.13

Multiply $- 1$ by $2$ .

$- 4 (\cot^{3} (θ) (- 2 \csc^{2} (θ)) + \csc^{2} (θ) (- 2 \cot (θ) \csc^{2} (θ))) + \frac{d}{d θ} [- 2 \csc^{4} (θ)]$

Step 4.2.14

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

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Step 4.2.14.1

Move $\csc^{2} (θ)$ .

$- 4 (\cot^{3} (θ) (- 2 \csc^{2} (θ)) + \csc^{2} (θ) \csc^{2} (θ) (- 2 \cot (θ))) + \frac{d}{d θ} [- 2 \csc^{4} (θ)]$

Step 4.2.14.2

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

$- 4 (\cot^{3} (θ) (- 2 \csc^{2} (θ)) + {\csc (θ)}^{2 + 2} (- 2 \cot (θ))) + \frac{d}{d θ} [- 2 \csc^{4} (θ)]$

Step 4.2.14.3

Add $2$ and $2$ .

$- 4 (\cot^{3} (θ) (- 2 \csc^{2} (θ)) + \csc^{4} (θ) (- 2 \cot (θ))) + \frac{d}{d θ} [- 2 \csc^{4} (θ)]$

Step 4.3

Evaluate $\frac{d}{d θ} [- 2 \csc^{4} (θ)]$ .

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Step 4.3.1

Since $- 2$ is constant with respect to $θ$ , the derivative of $- 2 \csc^{4} (θ)$ with respect to $θ$ is $- 2 \frac{d}{d θ} [\csc^{4} (θ)]$ .

$- 4 (\cot^{3} (θ) (- 2 \csc^{2} (θ)) + \csc^{4} (θ) (- 2 \cot (θ))) - 2 \frac{d}{d θ} [\csc^{4} (θ)]$

Step 4.3.2

Differentiate using the chain rule, which states that $\frac{d}{d θ} [f (g (θ))]$ is $f' (g (θ)) g' (θ)$ where $f (θ) = θ^{4}$ and $g (θ) = \csc (θ)$ .

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Step 4.3.2.1

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

$- 4 (\cot^{3} (θ) (- 2 \csc^{2} (θ)) + \csc^{4} (θ) (- 2 \cot (θ))) - 2 (\frac{d}{d u_{3}} [{u_{3}}^{4}] \frac{d}{d θ} [\csc (θ)])$

Step 4.3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{3}} [{u_{3}}^{n}]$ is $n {u_{3}}^{n - 1}$ where $n = 4$ .

$- 4 (\cot^{3} (θ) (- 2 \csc^{2} (θ)) + \csc^{4} (θ) (- 2 \cot (θ))) - 2 (4 {u_{3}}^{3} \frac{d}{d θ} [\csc (θ)])$

Step 4.3.2.3

Replace all occurrences of $u_{3}$ with $\csc (θ)$ .

$- 4 (\cot^{3} (θ) (- 2 \csc^{2} (θ)) + \csc^{4} (θ) (- 2 \cot (θ))) - 2 (4 \csc^{3} (θ) \frac{d}{d θ} [\csc (θ)])$

Step 4.3.3

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

$- 4 (\cot^{3} (θ) (- 2 \csc^{2} (θ)) + \csc^{4} (θ) (- 2 \cot (θ))) - 2 (4 \csc^{3} (θ) (- \csc (θ) \cot (θ)))$

Step 4.3.4

Multiply $- 1$ by $4$ .

$- 4 (\cot^{3} (θ) (- 2 \csc^{2} (θ)) + \csc^{4} (θ) (- 2 \cot (θ))) - 2 (- 4 \csc^{3} (θ) (\csc (θ) \cot (θ)))$

Step 4.3.5

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

$- 4 (\cot^{3} (θ) (- 2 \csc^{2} (θ)) + \csc^{4} (θ) (- 2 \cot (θ))) - 2 (- 4 (\csc^{1} (θ) \csc^{3} (θ)) \cot (θ))$

Step 4.3.6

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

$- 4 (\cot^{3} (θ) (- 2 \csc^{2} (θ)) + \csc^{4} (θ) (- 2 \cot (θ))) - 2 (- 4 {\csc (θ)}^{1 + 3} \cot (θ))$

Step 4.3.7

Add $1$ and $3$ .

$- 4 (\cot^{3} (θ) (- 2 \csc^{2} (θ)) + \csc^{4} (θ) (- 2 \cot (θ))) - 2 (- 4 \csc^{4} (θ) \cot (θ))$

Step 4.3.8

Multiply $- 4$ by $- 2$ .

$- 4 (\cot^{3} (θ) (- 2 \csc^{2} (θ)) + \csc^{4} (θ) (- 2 \cot (θ))) + 8 \csc^{4} (θ) \cot (θ)$

Step 4.4

Simplify.

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Step 4.4.1

Apply the distributive property.

$- 4 (\cot^{3} (θ) (- 2 \csc^{2} (θ))) - 4 (\csc^{4} (θ) (- 2 \cot (θ))) + 8 \csc^{4} (θ) \cot (θ)$

Step 4.4.2

Combine terms.

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Step 4.4.2.1

Multiply $- 2$ by $- 4$ .

$8 (\cot^{3} (θ) (\csc^{2} (θ))) - 4 (\csc^{4} (θ) (- 2 \cot (θ))) + 8 \csc^{4} (θ) \cot (θ)$

Step 4.4.2.2

Multiply $- 2$ by $- 4$ .

$8 \cot^{3} (θ) \csc^{2} (θ) + 8 (\csc^{4} (θ) (\cot (θ))) + 8 \csc^{4} (θ) \cot (θ)$

Step 4.4.2.3

Add $8 \csc^{4} (θ) \cot (θ)$ and $8 \csc^{4} (θ) \cot (θ)$ .

$f^{4} (θ) = 8 \cot^{3} (θ) \csc^{2} (θ) + 16 \csc^{4} (θ) \cot (θ)$