Calculus Examples

$7 \cot (x)$

Step 1

Find the first derivative.

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Step 1.1

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

$7 \frac{d}{d x} [\cot (x)]$

Step 1.2

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

$7 (- \csc^{2} (x))$

Step 1.3

Multiply $- 1$ by $7$ .

$f' (x) = - 7 \csc^{2} (x)$

Step 2

Find the second derivative.

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Step 2.1

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

$- 7 \frac{d}{d x} [\csc^{2} (x)]$

Step 2.2

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

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Step 2.2.1

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

$- 7 (\frac{d}{d u} [u^{2}] \frac{d}{d x} [\csc (x)])$

Step 2.2.2

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

$- 7 (2 u \frac{d}{d x} [\csc (x)])$

Step 2.2.3

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

$- 7 (2 \csc (x) \frac{d}{d x} [\csc (x)])$

Step 2.3

Multiply $2$ by $- 7$ .

$- 14 (\csc (x) \frac{d}{d x} [\csc (x)])$

Step 2.4

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

$- 14 \csc (x) (- \csc (x) \cot (x))$

Step 2.5

Multiply $- 1$ by $- 14$ .

$14 \csc (x) (\csc (x) \cot (x))$

Step 2.6

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

$14 (\csc^{1} (x) \csc (x)) \cot (x)$

Step 2.7

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

$14 (\csc^{1} (x) \csc^{1} (x)) \cot (x)$

Step 2.8

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

$14 {\csc (x)}^{1 + 1} \cot (x)$

Step 2.9

Add $1$ and $1$ .

$f'' (x) = 14 \csc^{2} (x) \cot (x)$

Step 3

Find the third derivative.

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Step 3.1

Since $14$ is constant with respect to $x$ , the derivative of $14 \csc^{2} (x) \cot (x)$ with respect to $x$ is $14 \frac{d}{d x} [\csc^{2} (x) \cot (x)]$ .

$14 \frac{d}{d x} [\csc^{2} (x) \cot (x)]$

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

$14 (\csc^{2} (x) \frac{d}{d x} [\cot (x)] + \cot (x) \frac{d}{d x} [\csc^{2} (x)])$

Step 3.3

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

$14 (\csc^{2} (x) (- \csc^{2} (x)) + \cot (x) \frac{d}{d x} [\csc^{2} (x)])$

Step 3.4

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

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Step 3.4.1

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

$14 (\csc^{2} (x) \csc^{2} (x) \cdot - 1 + \cot (x) \frac{d}{d x} [\csc^{2} (x)])$

Step 3.4.2

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

$14 ({\csc (x)}^{2 + 2} \cdot - 1 + \cot (x) \frac{d}{d x} [\csc^{2} (x)])$

Step 3.4.3

Add $2$ and $2$ .

$14 (\csc^{4} (x) \cdot - 1 + \cot (x) \frac{d}{d x} [\csc^{2} (x)])$

Step 3.5

Simplify the expression.

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Step 3.5.1

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

$14 (- 1 \cdot \csc^{4} (x) + \cot (x) \frac{d}{d x} [\csc^{2} (x)])$

Step 3.5.2

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

$14 (- \csc^{4} (x) + \cot (x) \frac{d}{d x} [\csc^{2} (x)])$

Step 3.6

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

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Step 3.6.1

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

$14 (- \csc^{4} (x) + \cot (x) (\frac{d}{d u} [u^{2}] \frac{d}{d x} [\csc (x)]))$

Step 3.6.2

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

$14 (- \csc^{4} (x) + \cot (x) (2 u \frac{d}{d x} [\csc (x)]))$

Step 3.6.3

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

$14 (- \csc^{4} (x) + \cot (x) (2 \csc (x) \frac{d}{d x} [\csc (x)]))$

Step 3.7

Move $2$ to the left of $\cot (x)$ .

$14 (- \csc^{4} (x) + 2 \cdot \cot (x) \csc (x) \frac{d}{d x} [\csc (x)])$

Step 3.8

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

$14 (- \csc^{4} (x) + 2 \cot (x) \csc (x) (- \csc (x) \cot (x)))$

Step 3.9

Multiply $- 1$ by $2$ .

$14 (- \csc^{4} (x) - 2 \cot (x) \csc (x) (\csc (x) \cot (x)))$

Step 3.10

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

$14 (- \csc^{4} (x) - 2 (\cot^{1} (x) \cot (x)) \csc (x) \csc (x))$

Step 3.11

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

$14 (- \csc^{4} (x) - 2 (\cot^{1} (x) \cot^{1} (x)) \csc (x) \csc (x))$

Step 3.12

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

$14 (- \csc^{4} (x) - 2 {\cot (x)}^{1 + 1} \csc (x) \csc (x))$

Step 3.13

Add $1$ and $1$ .

$14 (- \csc^{4} (x) - 2 \cot^{2} (x) \csc (x) \csc (x))$

Step 3.14

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

$14 (- \csc^{4} (x) - 2 \cot^{2} (x) (\csc^{1} (x) \csc (x)))$

Step 3.15

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

$14 (- \csc^{4} (x) - 2 \cot^{2} (x) (\csc^{1} (x) \csc^{1} (x)))$

Step 3.16

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

$14 (- \csc^{4} (x) - 2 \cot^{2} (x) {\csc (x)}^{1 + 1})$

Step 3.17

Add $1$ and $1$ .

$14 (- \csc^{4} (x) - 2 \cot^{2} (x) \csc^{2} (x))$

Step 3.18

Simplify.

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Step 3.18.1

Apply the distributive property.

$14 (- \csc^{4} (x)) + 14 (- 2 \cot^{2} (x) \csc^{2} (x))$

Step 3.18.2

Combine terms.

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Step 3.18.2.1

Multiply $- 1$ by $14$ .

$- 14 \csc^{4} (x) + 14 (- 2 \cot^{2} (x) \csc^{2} (x))$

Step 3.18.2.2

Multiply $- 2$ by $14$ .

$- 14 \csc^{4} (x) - 28 \cot^{2} (x) \csc^{2} (x)$

Step 3.18.3

Reorder terms.

$f''' (x) = - 28 \cot^{2} (x) \csc^{2} (x) - 14 \csc^{4} (x)$