If f(x) = sin x, what is the 99th derivative of

Hello,
Frankly my dear, who wants to compute the 99th derivative of sin(x)?

d/dx(sin(x))= cos(x)
d^2/dx^2 (sin(x))= d/dx( cos(x))=-sin(x)
d^3/dx^3 (sin(x))= -cos(x)
d^4/dx^4 (sin(x))=d/dx(-cos(x))=sinx.
In 100 successive differentiations you perform the foregoing sequence 100/4=25 times, and you end up with sin(x). Hence in the 99th step the derivative was -cos(x).

Hope it helps.