Question

find the 77th derivative of y=sin(3x)

Answer 1

It's a matter of pattern recognition.
\[\begin{array}{c|c}
n&y^{(n)}\\
\hline
0&\sin3x\\
1&3\cos3x\\
2&-3^2\sin3x\\
3&-3^3\cos3x\\
4&3^4\sin3x\\
\vdots&\vdots
\end{array}\]

Answer 2

As you can see, every two orders of the derivative, the sign of the expression changes. To figure out the sign of the 77th derivative, you need to determine where 78th term of the sequence,
\[\{1,1,-1,-1,1,1,-1,-1,1,1,\cdots\}\]
where the first term refers to \(n=0\), so the 78th term will refer to \(n=77\).

You should also be able to see that the exponent of the 3 is the same as the order of the derivative.

As for whether there is a sine or cosine, you can figure this out by noticing that every even-order derivative contains sine while every odd-order contains cosine.