Calculus Examples

$\frac{d^{119}}{d x^{119}} (\sin (x))$

Step 1

This derivative could not be completed using the chain rule. Mathway will use another method.

Step 2

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

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

Step 3

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

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

Step 4

Find the third derivative.

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Step 4.1

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 4.2

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

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

Step 5

Find the fourth derivative.

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Step 5.1

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 5.2

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

$- - \sin (x)$

Step 5.3

Multiply.

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Step 5.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 5.3.2

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

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

Step 6

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

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

Step 7

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

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

Step 8

Find the 7th derivative.

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Step 8.1

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 8.2

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

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

Step 9

Find the 8th derivative.

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Step 9.1

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 9.2

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

$- - \sin (x)$

Step 9.3

Multiply.

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Step 9.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 9.3.2

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

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

Step 10

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

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

Step 11

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

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

Step 12

Find the 11th derivative.

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Step 12.1

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 12.2

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

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

Step 13

Find the 12th derivative.

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Step 13.1

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 13.2

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

$- - \sin (x)$

Step 13.3

Multiply.

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Step 13.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 13.3.2

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

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

Step 14

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

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

Step 15

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

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

Step 16

Find the 15th derivative.

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Step 16.1

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 16.2

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

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

Step 17

Find the 16th derivative.

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Step 17.1

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 17.2

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

$- - \sin (x)$

Step 17.3

Multiply.

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Step 17.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 17.3.2

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

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

Step 18

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

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

Step 19

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

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

Step 20

Find the 19th derivative.

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Step 20.1

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 20.2

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

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

Step 21

Find the 20th derivative.

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Step 21.1

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 21.2

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

$- - \sin (x)$

Step 21.3

Multiply.

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Step 21.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 21.3.2

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

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

Step 22

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

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

Step 23

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

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

Step 24

Find the 23rd derivative.

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Step 24.1

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 24.2

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

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

Step 25

Find the 24th derivative.

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Step 25.1

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 25.2

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

$- - \sin (x)$

Step 25.3

Multiply.

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Step 25.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 25.3.2

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

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

Step 26

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

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

Step 27

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

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

Step 28

Find the 27th derivative.

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Step 28.1

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 28.2

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

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

Step 29

Find the 28th derivative.

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Step 29.1

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 29.2

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

$- - \sin (x)$

Step 29.3

Multiply.

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Step 29.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 29.3.2

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

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

Step 30

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

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

Step 31

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

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

Step 32

Find the 31st derivative.

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Step 32.1

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 32.2

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

$f^{31} (x) = - \cos (x)$

Step 33

Find the 32nd derivative.

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Step 33.1

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 33.2

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

$- - \sin (x)$

Step 33.3

Multiply.

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Step 33.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 33.3.2

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

$f^{32} (x) = \sin (x)$

Step 34

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

$f^{33} (x) = \cos (x)$

Step 35

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

$f^{34} (x) = - \sin (x)$

Step 36

Find the 35th derivative.

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Step 36.1

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 36.2

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

$f^{35} (x) = - \cos (x)$

Step 37

Find the 36th derivative.

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Step 37.1

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 37.2

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

$- - \sin (x)$

Step 37.3

Multiply.

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Step 37.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 37.3.2

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

$f^{36} (x) = \sin (x)$

Step 38

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

$f^{37} (x) = \cos (x)$

Step 39

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

$f^{38} (x) = - \sin (x)$

Step 40

Find the 39th derivative.

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Step 40.1

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 40.2

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

$f^{39} (x) = - \cos (x)$

Step 41

Find the 40th derivative.

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Step 41.1

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 41.2

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

$- - \sin (x)$

Step 41.3

Multiply.

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Step 41.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 41.3.2

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

$f^{40} (x) = \sin (x)$

Step 42

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

$f^{41} (x) = \cos (x)$

Step 43

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

$f^{42} (x) = - \sin (x)$

Step 44

Find the 43rd derivative.

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Step 44.1

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 44.2

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

$f^{43} (x) = - \cos (x)$

Step 45

Find the 44th derivative.

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Step 45.1

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 45.2

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

$- - \sin (x)$

Step 45.3

Multiply.

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Step 45.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 45.3.2

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

$f^{44} (x) = \sin (x)$

Step 46

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

$f^{45} (x) = \cos (x)$

Step 47

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

$f^{46} (x) = - \sin (x)$

Step 48

Find the 47th derivative.

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Step 48.1

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 48.2

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

$f^{47} (x) = - \cos (x)$

Step 49

Find the 48th derivative.

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Step 49.1

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 49.2

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

$- - \sin (x)$

Step 49.3

Multiply.

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Step 49.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 49.3.2

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

$f^{48} (x) = \sin (x)$

Step 50

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

$f^{49} (x) = \cos (x)$

Step 51

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

$f^{50} (x) = - \sin (x)$

Step 52

Find the 51st derivative.

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Step 52.1

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 52.2

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

$f^{51} (x) = - \cos (x)$

Step 53

Find the 52nd derivative.

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Step 53.1

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 53.2

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

$- - \sin (x)$

Step 53.3

Multiply.

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Step 53.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 53.3.2

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

$f^{52} (x) = \sin (x)$

Step 54

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

$f^{53} (x) = \cos (x)$

Step 55

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

$f^{54} (x) = - \sin (x)$

Step 56

Find the 55th derivative.

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Step 56.1

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 56.2

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

$f^{55} (x) = - \cos (x)$

Step 57

Find the 56th derivative.

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Step 57.1

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 57.2

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

$- - \sin (x)$

Step 57.3

Multiply.

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Step 57.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 57.3.2

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

$f^{56} (x) = \sin (x)$

Step 58

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

$f^{57} (x) = \cos (x)$

Step 59

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

$f^{58} (x) = - \sin (x)$

Step 60

Find the 59th derivative.

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Step 60.1

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 60.2

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

$f^{59} (x) = - \cos (x)$

Step 61

Find the 60th derivative.

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Step 61.1

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 61.2

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

$- - \sin (x)$

Step 61.3

Multiply.

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Step 61.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 61.3.2

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

$f^{60} (x) = \sin (x)$

Step 62

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

$f^{61} (x) = \cos (x)$

Step 63

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

$f^{62} (x) = - \sin (x)$

Step 64

Find the 63rd derivative.

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Step 64.1

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 64.2

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

$f^{63} (x) = - \cos (x)$

Step 65

Find the 64th derivative.

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Step 65.1

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 65.2

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

$- - \sin (x)$

Step 65.3

Multiply.

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Step 65.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 65.3.2

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

$f^{64} (x) = \sin (x)$

Step 66

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

$f^{65} (x) = \cos (x)$

Step 67

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

$f^{66} (x) = - \sin (x)$

Step 68

Find the 67th derivative.

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Step 68.1

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 68.2

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

$f^{67} (x) = - \cos (x)$

Step 69

Find the 68th derivative.

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Step 69.1

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 69.2

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

$- - \sin (x)$

Step 69.3

Multiply.

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Step 69.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 69.3.2

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

$f^{68} (x) = \sin (x)$

Step 70

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

$f^{69} (x) = \cos (x)$

Step 71

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

$f^{70} (x) = - \sin (x)$

Step 72

Find the 71st derivative.

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Step 72.1

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 72.2

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

$f^{71} (x) = - \cos (x)$

Step 73

Find the 72nd derivative.

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Step 73.1

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 73.2

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

$- - \sin (x)$

Step 73.3

Multiply.

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Step 73.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 73.3.2

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

$f^{72} (x) = \sin (x)$

Step 74

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

$f^{73} (x) = \cos (x)$

Step 75

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

$f^{74} (x) = - \sin (x)$

Step 76

Find the 75th derivative.

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Step 76.1

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 76.2

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

$f^{75} (x) = - \cos (x)$

Step 77

Find the 76th derivative.

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Step 77.1

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 77.2

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

$- - \sin (x)$

Step 77.3

Multiply.

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Step 77.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 77.3.2

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

$f^{76} (x) = \sin (x)$

Step 78

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

$f^{77} (x) = \cos (x)$

Step 79

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

$f^{78} (x) = - \sin (x)$

Step 80

Find the 79th derivative.

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Step 80.1

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 80.2

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

$f^{79} (x) = - \cos (x)$

Step 81

Find the 80th derivative.

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Step 81.1

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 81.2

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

$- - \sin (x)$

Step 81.3

Multiply.

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Step 81.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 81.3.2

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

$f^{80} (x) = \sin (x)$

Step 82

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

$f^{81} (x) = \cos (x)$

Step 83

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

$f^{82} (x) = - \sin (x)$

Step 84

Find the 83rd derivative.

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Step 84.1

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 84.2

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

$f^{83} (x) = - \cos (x)$

Step 85

Find the 84th derivative.

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Step 85.1

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 85.2

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

$- - \sin (x)$

Step 85.3

Multiply.

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Step 85.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 85.3.2

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

$f^{84} (x) = \sin (x)$

Step 86

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

$f^{85} (x) = \cos (x)$

Step 87

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

$f^{86} (x) = - \sin (x)$

Step 88

Find the 87th derivative.

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Step 88.1

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 88.2

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

$f^{87} (x) = - \cos (x)$

Step 89

Find the 88th derivative.

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Step 89.1

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 89.2

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

$- - \sin (x)$

Step 89.3

Multiply.

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Step 89.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 89.3.2

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

$f^{88} (x) = \sin (x)$

Step 90

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

$f^{89} (x) = \cos (x)$

Step 91

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

$f^{90} (x) = - \sin (x)$

Step 92

Find the 91st derivative.

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Step 92.1

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 92.2

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

$f^{91} (x) = - \cos (x)$

Step 93

Find the 92nd derivative.

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Step 93.1

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 93.2

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

$- - \sin (x)$

Step 93.3

Multiply.

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Step 93.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 93.3.2

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

$f^{92} (x) = \sin (x)$

Step 94

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

$f^{93} (x) = \cos (x)$

Step 95

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

$f^{94} (x) = - \sin (x)$

Step 96

Find the 95th derivative.

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Step 96.1

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 96.2

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

$f^{95} (x) = - \cos (x)$

Step 97

Find the 96th derivative.

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Step 97.1

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 97.2

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

$- - \sin (x)$

Step 97.3

Multiply.

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Step 97.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 97.3.2

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

$f^{96} (x) = \sin (x)$

Step 98

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

$f^{97} (x) = \cos (x)$

Step 99

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

$f^{98} (x) = - \sin (x)$

Step 100

Find the 99th derivative.

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Step 100.1

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 100.2

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

$f^{99} (x) = - \cos (x)$

Step 101

Find the 100th derivative.

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Step 101.1

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 101.2

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

$- - \sin (x)$

Step 101.3

Multiply.

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Step 101.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 101.3.2

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

$f^{100} (x) = \sin (x)$

Step 102

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

$f^{101} (x) = \cos (x)$

Step 103

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

$f^{102} (x) = - \sin (x)$

Step 104

Find the 103rd derivative.

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Step 104.1

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 104.2

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

$f^{103} (x) = - \cos (x)$

Step 105

Find the 104th derivative.

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Step 105.1

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 105.2

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

$- - \sin (x)$

Step 105.3

Multiply.

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Step 105.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 105.3.2

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

$f^{104} (x) = \sin (x)$

Step 106

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

$f^{105} (x) = \cos (x)$

Step 107

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

$f^{106} (x) = - \sin (x)$

Step 108

Find the 107th derivative.

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Step 108.1

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 108.2

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

$f^{107} (x) = - \cos (x)$

Step 109

Find the 108th derivative.

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Step 109.1

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 109.2

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

$- - \sin (x)$

Step 109.3

Multiply.

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Step 109.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 109.3.2

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

$f^{108} (x) = \sin (x)$

Step 110

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

$f^{109} (x) = \cos (x)$

Step 111

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

$f^{110} (x) = - \sin (x)$

Step 112

Find the 111th derivative.

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Step 112.1

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 112.2

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

$f^{111} (x) = - \cos (x)$

Step 113

Find the 112th derivative.

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Step 113.1

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 113.2

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

$- - \sin (x)$

Step 113.3

Multiply.

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Step 113.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 113.3.2

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

$f^{112} (x) = \sin (x)$

Step 114

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

$f^{113} (x) = \cos (x)$

Step 115

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

$f^{114} (x) = - \sin (x)$

Step 116

Find the 115th derivative.

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Step 116.1

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 116.2

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

$f^{115} (x) = - \cos (x)$

Step 117

Find the 116th derivative.

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Step 117.1

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 117.2

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

$- - \sin (x)$

Step 117.3

Multiply.

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Step 117.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 117.3.2

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

$f^{116} (x) = \sin (x)$

Step 118

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

$f^{117} (x) = \cos (x)$

Step 119

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

$f^{118} (x) = - \sin (x)$

Step 120

Find the 119th derivative.

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Step 120.1

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 120.2

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

$f^{119} (x) = - \cos (x)$