Calculus Examples

$\frac{d^{90}}{d x^{90}} \cdot (\sin (x))$

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 $- \sin (x)$ .

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

Step 31

Find the 31st derivative.

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Step 31.1

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 31.2

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

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

Step 32

Find the 32nd derivative.

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Step 32.1

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 32.2

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

$- - \sin (x)$

Step 32.3

Multiply.

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Step 32.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 32.3.2

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

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

Step 33

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

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

Step 34

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

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

Step 35

Find the 35th derivative.

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Step 35.1

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 35.2

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

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

Step 36

Find the 36th derivative.

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Step 36.1

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 36.2

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

$- - \sin (x)$

Step 36.3

Multiply.

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Step 36.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 36.3.2

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

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

Step 37

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

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

Step 38

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

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

Step 39

Find the 39th derivative.

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Step 39.1

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 39.2

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

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

Step 40

Find the 40th derivative.

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Step 40.1

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 40.2

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

$- - \sin (x)$

Step 40.3

Multiply.

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Step 40.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 40.3.2

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

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

Step 41

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

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

Step 42

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

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

Step 43

Find the 43rd derivative.

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Step 43.1

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 43.2

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

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

Step 44

Find the 44th derivative.

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Step 44.1

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 44.2

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

$- - \sin (x)$

Step 44.3

Multiply.

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Step 44.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 44.3.2

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

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

Step 45

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

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

Step 46

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

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

Step 47

Find the 47th derivative.

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Step 47.1

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 47.2

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

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

Step 48

Find the 48th derivative.

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Step 48.1

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 48.2

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

$- - \sin (x)$

Step 48.3

Multiply.

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Step 48.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 48.3.2

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

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

Step 49

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

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

Step 50

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

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

Step 51

Find the 51st derivative.

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Step 51.1

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 51.2

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

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

Step 52

Find the 52nd derivative.

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Step 52.1

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 52.2

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

$- - \sin (x)$

Step 52.3

Multiply.

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Step 52.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 52.3.2

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

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

Step 53

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

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

Step 54

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

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

Step 55

Find the 55th derivative.

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Step 55.1

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 55.2

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

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

Step 56

Find the 56th derivative.

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Step 56.1

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 56.2

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

$- - \sin (x)$

Step 56.3

Multiply.

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Step 56.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 56.3.2

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

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

Step 57

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

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

Step 58

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

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

Step 59

Find the 59th derivative.

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Step 59.1

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 59.2

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

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

Step 60

Find the 60th derivative.

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Step 60.1

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 60.2

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

$- - \sin (x)$

Step 60.3

Multiply.

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Step 60.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 60.3.2

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

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

Step 61

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

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

Step 62

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

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

Step 63

Find the 63rd derivative.

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Step 63.1

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 63.2

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

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

Step 64

Find the 64th derivative.

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Step 64.1

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 64.2

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

$- - \sin (x)$

Step 64.3

Multiply.

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Step 64.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 64.3.2

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

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

Step 65

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

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

Step 66

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

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

Step 67

Find the 67th derivative.

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Step 67.1

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 67.2

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

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

Step 68

Find the 68th derivative.

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Step 68.1

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 68.2

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

$- - \sin (x)$

Step 68.3

Multiply.

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Step 68.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 68.3.2

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

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

Step 69

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

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

Step 70

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

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

Step 71

Find the 71st derivative.

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Step 71.1

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 71.2

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

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

Step 72

Find the 72nd derivative.

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Step 72.1

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 72.2

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

$- - \sin (x)$

Step 72.3

Multiply.

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Step 72.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 72.3.2

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

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

Step 73

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

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

Step 74

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

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

Step 75

Find the 75th derivative.

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Step 75.1

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 75.2

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

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

Step 76

Find the 76th derivative.

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Step 76.1

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 76.2

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

$- - \sin (x)$

Step 76.3

Multiply.

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Step 76.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 76.3.2

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

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

Step 77

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

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

Step 78

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

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

Step 79

Find the 79th derivative.

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Step 79.1

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 79.2

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

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

Step 80

Find the 80th derivative.

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Step 80.1

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 80.2

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

$- - \sin (x)$

Step 80.3

Multiply.

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Step 80.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 80.3.2

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

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

Step 81

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

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

Step 82

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

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

Step 83

Find the 83rd derivative.

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Step 83.1

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 83.2

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

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

Step 84

Find the 84th derivative.

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Step 84.1

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 84.2

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

$- - \sin (x)$

Step 84.3

Multiply.

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Step 84.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 84.3.2

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

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

Step 85

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

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

Step 86

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

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

Step 87

Find the 87th derivative.

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Step 87.1

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 87.2

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

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

Step 88

Find the 88th derivative.

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Step 88.1

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 88.2

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

$- - \sin (x)$

Step 88.3

Multiply.

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Step 88.3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 88.3.2

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

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

Step 89

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

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

Step 90

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

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