What rule can I use to compute $\frac{d^{107}}{dx^{107}} \sin x$? Did I miss something in my calculus class? I don't remember anything concerning this type of problem: 

Compute $$\frac{d^{107}}{dx^{107}} \sin x.$$

So what is the rule here?

Notice, cyclic order of derivative of $\sin x$ is $4 $ hence the power (order of derivative) should be divided by $4$ & remainder is checked as follows
$$\begin{cases}\frac{d}{dx}(\sin x)=\cos x\ \text{(remainder}=1)\\ 
\frac{d^2}{dx^2}(\sin x)=-\sin x \ \text{(remainder}=2)\\
\frac{d^3}{dx^3}(\sin x)=-\cos x\ \text{(remainder}=3)\\
\frac{d^4}{dx^4}(\sin x)=\sin x\ \text{(remainder}=0)\\
\end{cases}$$
Now, we divide power (order) of derivative $107$ by $4$ to find remainder as follows $$107=4\times 26+3$$
In this case, we have $\text{(remainder}=3)$ hence case (3) is the answer
$$\bbox[5px, border:2px solid #C0A000]{\color{red}{\frac{d^{107}}{dx^{107}}(\sin x)=-cos x}}$$
