Algebra Examples

$\frac{d^{30}}{d x^{30}} (\sin (x))$

Step 1

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

$f' (x) = \cos (x)$

Step 2

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

$f'' (x) = - \sin (x)$

Step 3

Find the third derivative.

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Step 3.1

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

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

Step 3.2

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

$f''' (x) = - \cos (x)$

Step 4

Find the fourth derivative.

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Step 4.1

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

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

Step 4.2

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

$- - \sin (x)$

Step 4.3

Multiply.

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Step 4.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 4.3.2

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

$f^{4} (x) = \sin (x)$

Step 5

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

$f^{5} (x) = \cos (x)$

Step 6

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

$f^{6} (x) = - \sin (x)$

Step 7

Find the 7th derivative.

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Step 7.1

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

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

Step 7.2

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

$f^{7} (x) = - \cos (x)$

Step 8

Find the 8th derivative.

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Step 8.1

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

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

Step 8.2

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

$- - \sin (x)$

Step 8.3

Multiply.

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Step 8.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 8.3.2

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

$f^{8} (x) = \sin (x)$

Step 9

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

$f^{9} (x) = \cos (x)$

Step 10

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

$f^{10} (x) = - \sin (x)$

Step 11

Find the 11th derivative.

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Step 11.1

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

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

Step 11.2

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

$f^{11} (x) = - \cos (x)$

Step 12

Find the 12th derivative.

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Step 12.1

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

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

Step 12.2

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

$- - \sin (x)$

Step 12.3

Multiply.

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Step 12.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 12.3.2

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

$f^{12} (x) = \sin (x)$

Step 13

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

$f^{13} (x) = \cos (x)$

Step 14

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

$f^{14} (x) = - \sin (x)$

Step 15

Find the 15th derivative.

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Step 15.1

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

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

Step 15.2

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

$f^{15} (x) = - \cos (x)$

Step 16

Find the 16th derivative.

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Step 16.1

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

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

Step 16.2

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

$- - \sin (x)$

Step 16.3

Multiply.

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Step 16.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 16.3.2

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

$f^{16} (x) = \sin (x)$

Step 17

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

$f^{17} (x) = \cos (x)$

Step 18

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

$f^{18} (x) = - \sin (x)$

Step 19

Find the 19th derivative.

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Step 19.1

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

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

Step 19.2

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

$f^{19} (x) = - \cos (x)$

Step 20

Find the 20th derivative.

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Step 20.1

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

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

Step 20.2

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

$- - \sin (x)$

Step 20.3

Multiply.

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Step 20.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 20.3.2

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

$f^{20} (x) = \sin (x)$

Step 21

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

$f^{21} (x) = \cos (x)$

Step 22

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

$f^{22} (x) = - \sin (x)$

Step 23

Find the 23rd derivative.

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Step 23.1

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

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

Step 23.2

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

$f^{23} (x) = - \cos (x)$

Step 24

Find the 24th derivative.

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Step 24.1

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

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

Step 24.2

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

$- - \sin (x)$

Step 24.3

Multiply.

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Step 24.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 24.3.2

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

$f^{24} (x) = \sin (x)$

Step 25

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

$f^{25} (x) = \cos (x)$

Step 26

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

$f^{26} (x) = - \sin (x)$

Step 27

Find the 27th derivative.

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Step 27.1

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

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

Step 27.2

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

$f^{27} (x) = - \cos (x)$

Step 28

Find the 28th derivative.

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Step 28.1

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

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

Step 28.2

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

$- - \sin (x)$

Step 28.3

Multiply.

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Step 28.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 28.3.2

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

$f^{28} (x) = \sin (x)$

Step 29

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

$f^{29} (x) = \cos (x)$

Step 30

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

$f^{30} (x) = - \sin (x)$