To find [tex] \frac{d^{103}}{dx^{103}} \left(\sin{(x)}\right)[/tex], we find the first few derivatives and observe the pattern that occurs.
[tex] \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)}[/tex]
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,
[tex]\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)}[/tex]
Therefore,
[tex]\frac{d^{103}}{dx^{103}} \left(\sin{(x)}\right)=-\cos{(x)}[/tex].
