Calculus Examples

$y = 9 \csc (x)$

Step 1

Find the first derivative.

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Step 1.1

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

$9 \frac{d}{d x} [\csc (x)]$

Step 1.2

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

$9 (- \csc (x) \cot (x))$

Step 1.3

Simplify the expression.

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Step 1.3.1

Multiply $- 1$ by $9$ .

$- 9 (\csc (x) \cot (x))$

Step 1.3.2

Reorder the factors of $- 9 \csc (x) \cot (x)$ .

$f' (x) = - 9 \cot (x) \csc (x)$

Step 2

Find the second derivative.

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Step 2.1

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

$- 9 \frac{d}{d x} [\cot (x) \csc (x)]$

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 $g (x) = \csc (x)$ .

$- 9 (\cot (x) \frac{d}{d x} [\csc (x)] + \csc (x) \frac{d}{d x} [\cot (x)])$

Step 2.3

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

$- 9 (\cot (x) (- \csc (x) \cot (x)) + \csc (x) \frac{d}{d x} [\cot (x)])$

Step 2.4

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

$- 9 (- \csc (x) (\cot^{1} (x) \cot (x)) + \csc (x) \frac{d}{d x} [\cot (x)])$

Step 2.5

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

$- 9 (- \csc (x) (\cot^{1} (x) \cot^{1} (x)) + \csc (x) \frac{d}{d x} [\cot (x)])$

Step 2.6

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

$- 9 (- \csc (x) {\cot (x)}^{1 + 1} + \csc (x) \frac{d}{d x} [\cot (x)])$

Step 2.7

Add $1$ and $1$ .

$- 9 (- \csc (x) \cot^{2} (x) + \csc (x) \frac{d}{d x} [\cot (x)])$

Step 2.8

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

$- 9 (- \csc (x) \cot^{2} (x) + \csc (x) (- \csc^{2} (x)))$

Step 2.9

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

$- 9 (- \csc (x) \cot^{2} (x) - (\csc^{1} (x) \csc^{2} (x)))$

Step 2.10

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

$- 9 (- \csc (x) \cot^{2} (x) - {\csc (x)}^{1 + 2})$

Step 2.11

Add $1$ and $2$ .

$- 9 (- \csc (x) \cot^{2} (x) - \csc^{3} (x))$

Step 2.12

Simplify.

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Step 2.12.1

Apply the distributive property.

$- 9 (- \csc (x) \cot^{2} (x)) - 9 (- \csc^{3} (x))$

Step 2.12.2

Combine terms.

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Step 2.12.2.1

Multiply $- 1$ by $- 9$ .

$9 (\csc (x) \cot^{2} (x)) - 9 (- \csc^{3} (x))$

Step 2.12.2.2

Multiply $- 1$ by $- 9$ .

$9 \csc (x) \cot^{2} (x) + 9 \csc^{3} (x)$

Step 2.12.3

Reorder terms.

$f'' (x) = 9 \cot^{2} (x) \csc (x) + 9 \csc^{3} (x)$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $9 \cot^{2} (x) \csc (x) + 9 \csc^{3} (x)$ with respect to $x$ is $\frac{d}{d x} [9 \cot^{2} (x) \csc (x)] + \frac{d}{d x} [9 \csc^{3} (x)]$ .

$\frac{d}{d x} [9 \cot^{2} (x) \csc (x)] + \frac{d}{d x} [9 \csc^{3} (x)]$

Step 3.2

Evaluate $\frac{d}{d x} [9 \cot^{2} (x) \csc (x)]$ .

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Step 3.2.1

Since $9$ is constant with respect to $x$ , the derivative of $9 \cot^{2} (x) \csc (x)$ with respect to $x$ is $9 \frac{d}{d x} [\cot^{2} (x) \csc (x)]$ .

$9 \frac{d}{d x} [\cot^{2} (x) \csc (x)] + \frac{d}{d x} [9 \csc^{3} (x)]$

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

$9 (\cot^{2} (x) \frac{d}{d x} [\csc (x)] + \csc (x) \frac{d}{d x} [\cot^{2} (x)]) + \frac{d}{d x} [9 \csc^{3} (x)]$

Step 3.2.3

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

$9 (\cot^{2} (x) (- \csc (x) \cot (x)) + \csc (x) \frac{d}{d x} [\cot^{2} (x)]) + \frac{d}{d x} [9 \csc^{3} (x)]$

Step 3.2.4

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

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Step 3.2.4.1

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

$9 (\cot^{2} (x) (- \csc (x) \cot (x)) + \csc (x) (\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [\cot (x)])) + \frac{d}{d x} [9 \csc^{3} (x)]$

Step 3.2.4.2

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

$9 (\cot^{2} (x) (- \csc (x) \cot (x)) + \csc (x) (2 u_{1} \frac{d}{d x} [\cot (x)])) + \frac{d}{d x} [9 \csc^{3} (x)]$

Step 3.2.4.3

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

$9 (\cot^{2} (x) (- \csc (x) \cot (x)) + \csc (x) (2 \cot (x) \frac{d}{d x} [\cot (x)])) + \frac{d}{d x} [9 \csc^{3} (x)]$

Step 3.2.5

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

$9 (\cot^{2} (x) (- \csc (x) \cot (x)) + \csc (x) (2 \cot (x) (- \csc^{2} (x)))) + \frac{d}{d x} [9 \csc^{3} (x)]$

Step 3.2.6

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

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Step 3.2.6.1

Move $\cot (x)$ .

$9 (\cot (x) \cot^{2} (x) (- \csc (x)) + \csc (x) (2 \cot (x) (- \csc^{2} (x)))) + \frac{d}{d x} [9 \csc^{3} (x)]$

Step 3.2.6.2

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

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Step 3.2.6.2.1

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

$9 (\cot^{1} (x) \cot^{2} (x) (- \csc (x)) + \csc (x) (2 \cot (x) (- \csc^{2} (x)))) + \frac{d}{d x} [9 \csc^{3} (x)]$

Step 3.2.6.2.2

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

$9 ({\cot (x)}^{1 + 2} (- \csc (x)) + \csc (x) (2 \cot (x) (- \csc^{2} (x)))) + \frac{d}{d x} [9 \csc^{3} (x)]$

Step 3.2.6.3

Add $1$ and $2$ .

$9 (\cot^{3} (x) (- \csc (x)) + \csc (x) (2 \cot (x) (- \csc^{2} (x)))) + \frac{d}{d x} [9 \csc^{3} (x)]$

Step 3.2.7

Multiply $- 1$ by $2$ .

$9 (\cot^{3} (x) (- \csc (x)) + \csc (x) (- 2 \cot (x) \csc^{2} (x))) + \frac{d}{d x} [9 \csc^{3} (x)]$

Step 3.2.8

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

$9 (\cot^{3} (x) (- \csc (x)) - 2 \cot (x) (\csc^{1} (x) \csc^{2} (x))) + \frac{d}{d x} [9 \csc^{3} (x)]$

Step 3.2.9

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

$9 (\cot^{3} (x) (- \csc (x)) - 2 \cot (x) {\csc (x)}^{1 + 2}) + \frac{d}{d x} [9 \csc^{3} (x)]$

Step 3.2.10

Add $1$ and $2$ .

$9 (\cot^{3} (x) (- \csc (x)) - 2 \cot (x) \csc^{3} (x)) + \frac{d}{d x} [9 \csc^{3} (x)]$

Step 3.3

Evaluate $\frac{d}{d x} [9 \csc^{3} (x)]$ .

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Step 3.3.1

Since $9$ is constant with respect to $x$ , the derivative of $9 \csc^{3} (x)$ with respect to $x$ is $9 \frac{d}{d x} [\csc^{3} (x)]$ .

$9 (\cot^{3} (x) (- \csc (x)) - 2 \cot (x) \csc^{3} (x)) + 9 \frac{d}{d x} [\csc^{3} (x)]$

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

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Step 3.3.2.1

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

$9 (\cot^{3} (x) (- \csc (x)) - 2 \cot (x) \csc^{3} (x)) + 9 (\frac{d}{d u_{2}} [{u_{2}}^{3}] \frac{d}{d x} [\csc (x)])$

Step 3.3.2.2

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

$9 (\cot^{3} (x) (- \csc (x)) - 2 \cot (x) \csc^{3} (x)) + 9 (3 {u_{2}}^{2} \frac{d}{d x} [\csc (x)])$

Step 3.3.2.3

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

$9 (\cot^{3} (x) (- \csc (x)) - 2 \cot (x) \csc^{3} (x)) + 9 (3 \csc^{2} (x) \frac{d}{d x} [\csc (x)])$

Step 3.3.3

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

$9 (\cot^{3} (x) (- \csc (x)) - 2 \cot (x) \csc^{3} (x)) + 9 (3 \csc^{2} (x) (- \csc (x) \cot (x)))$

Step 3.3.4

Multiply $- 1$ by $3$ .

$9 (\cot^{3} (x) (- \csc (x)) - 2 \cot (x) \csc^{3} (x)) + 9 (- 3 \csc^{2} (x) (\csc (x) \cot (x)))$

Step 3.3.5

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

$9 (\cot^{3} (x) (- \csc (x)) - 2 \cot (x) \csc^{3} (x)) + 9 (- 3 (\csc^{1} (x) \csc^{2} (x)) \cot (x))$

Step 3.3.6

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

$9 (\cot^{3} (x) (- \csc (x)) - 2 \cot (x) \csc^{3} (x)) + 9 (- 3 {\csc (x)}^{1 + 2} \cot (x))$

Step 3.3.7

Add $1$ and $2$ .

$9 (\cot^{3} (x) (- \csc (x)) - 2 \cot (x) \csc^{3} (x)) + 9 (- 3 \csc^{3} (x) \cot (x))$

Step 3.3.8

Multiply $- 3$ by $9$ .

$9 (\cot^{3} (x) (- \csc (x)) - 2 \cot (x) \csc^{3} (x)) - 27 \csc^{3} (x) \cot (x)$

Step 3.4

Simplify.

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Step 3.4.1

Apply the distributive property.

$9 (\cot^{3} (x) (- \csc (x))) + 9 (- 2 \cot (x) \csc^{3} (x)) - 27 \csc^{3} (x) \cot (x)$

Step 3.4.2

Combine terms.

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Step 3.4.2.1

Multiply $- 1$ by $9$ .

$- 9 (\cot^{3} (x) (\csc (x))) + 9 (- 2 \cot (x) \csc^{3} (x)) - 27 \csc^{3} (x) \cot (x)$

Step 3.4.2.2

Multiply $- 2$ by $9$ .

$- 9 \cot^{3} (x) \csc (x) - 18 (\cot (x) \csc^{3} (x)) - 27 \csc^{3} (x) \cot (x)$

Step 3.4.2.3

Reorder the factors of $- 18 \cot (x) \csc^{3} (x)$ .

$- 9 \cot^{3} (x) \csc (x) - 18 \csc^{3} (x) \cot (x) - 27 \csc^{3} (x) \cot (x)$

Step 3.4.2.4

Subtract $27 \csc^{3} (x) \cot (x)$ from $- 18 \csc^{3} (x) \cot (x)$ .

$f''' (x) = - 9 \cot^{3} (x) \csc (x) - 45 \csc^{3} (x) \cot (x)$