To find
![\frac{d^{103}}{dx^{103}} \left(\sin{(x)}\right)](https://tex.z-dn.net/?f=%20%5Cfrac%7Bd%5E%7B103%7D%7D%7Bdx%5E%7B103%7D%7D%20%5Cleft%28%5Csin%7B%28x%29%7D%5Cright%29)
, we find the first few derivatives and observe the pattern that occurs.
![\frac{d}{dx} (\sin{(x)})=\cos{(x)} \\ \\ \frac{d^2}{dx^2} (\sin{(x)})= \frac{d}{dx} (\cos{(x)})=-\sin{(x)} \\ \\ \frac{d^3}{dx^3} (\sin{(x)})= -\frac{d}{dx} (\sin{(x)})=-\cos{(x)} \\ \\ \frac{d^4}{dx^4} (\sin{(x)})= -\frac{d}{dx} (\cos{(x)})=-(-\sin{(x)})=\sin{(x)} \\ \\ \frac{d^5}{dx^5} (\sin{(x)})= \frac{d}{dx} (\sin{(x)})=\cos{(x)}](https://tex.z-dn.net/?f=%20%5Cfrac%7Bd%7D%7Bdx%7D%20%28%5Csin%7B%28x%29%7D%29%3D%5Ccos%7B%28x%29%7D%20%5C%5C%20%20%5C%5C%20%20%5Cfrac%7Bd%5E2%7D%7Bdx%5E2%7D%20%28%5Csin%7B%28x%29%7D%29%3D%20%5Cfrac%7Bd%7D%7Bdx%7D%20%28%5Ccos%7B%28x%29%7D%29%3D-%5Csin%7B%28x%29%7D%20%5C%5C%20%20%5C%5C%20%5Cfrac%7Bd%5E3%7D%7Bdx%5E3%7D%20%28%5Csin%7B%28x%29%7D%29%3D%20-%5Cfrac%7Bd%7D%7Bdx%7D%20%28%5Csin%7B%28x%29%7D%29%3D-%5Ccos%7B%28x%29%7D%20%5C%5C%20%20%5C%5C%20%5Cfrac%7Bd%5E4%7D%7Bdx%5E4%7D%20%28%5Csin%7B%28x%29%7D%29%3D%20-%5Cfrac%7Bd%7D%7Bdx%7D%20%28%5Ccos%7B%28x%29%7D%29%3D-%28-%5Csin%7B%28x%29%7D%29%3D%5Csin%7B%28x%29%7D%20%5C%5C%20%20%5C%5C%20%5Cfrac%7Bd%5E5%7D%7Bdx%5E5%7D%20%28%5Csin%7B%28x%29%7D%29%3D%20%20%5Cfrac%7Bd%7D%7Bdx%7D%20%28%5Csin%7B%28x%29%7D%29%3D%5Ccos%7B%28x%29%7D)
As can be seen above, it can be seen that the continuos derivative of sin (x) is a sequence which repeats after every four terms.
Thus,
![\frac{d^{103}}{dx^{103}} \left(\sin{(x)}\right)= \frac{d^{4(25)+3}}{dx^{4(25)+3}} \left(\sin{(x)}\right) \\ \\ = \frac{d^3}{dx^3} \left(\sin{(x)}\right)=-\cos{(x)}](https://tex.z-dn.net/?f=%5Cfrac%7Bd%5E%7B103%7D%7D%7Bdx%5E%7B103%7D%7D%20%5Cleft%28%5Csin%7B%28x%29%7D%5Cright%29%3D%20%5Cfrac%7Bd%5E%7B4%2825%29%2B3%7D%7D%7Bdx%5E%7B4%2825%29%2B3%7D%7D%20%5Cleft%28%5Csin%7B%28x%29%7D%5Cright%29%20%5C%5C%20%20%5C%5C%20%3D%20%5Cfrac%7Bd%5E3%7D%7Bdx%5E3%7D%20%5Cleft%28%5Csin%7B%28x%29%7D%5Cright%29%3D-%5Ccos%7B%28x%29%7D)
Therefore,
![\frac{d^{103}}{dx^{103}} \left(\sin{(x)}\right)=-\cos{(x)}](https://tex.z-dn.net/?f=%5Cfrac%7Bd%5E%7B103%7D%7D%7Bdx%5E%7B103%7D%7D%20%5Cleft%28%5Csin%7B%28x%29%7D%5Cright%29%3D-%5Ccos%7B%28x%29%7D)
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Answer:
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Step-by-step explanation:
- 2a -12
Answer:
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Step-by-step explanation: