Question

how do you find the f^34(x) derivative? of cosx in particular?

Answer 1

the pattern goes
f^0=cosx
f'=-sinx
f''=-cosx
f'''=sinx
f^4=cosx=f^0
So it repeats every 4 derivatives
So if you want f^34 think of it as f^(34/4)=f^(8+2/4)
So only the 2/4 is important. The numerator represents what number of derivatives you effectively need to take. We then here only need to take the second derivative of cosx
therefor f^34(x)=-cosx