Question #d0c3c

1 Answer
Sep 27, 2017

-sin x


Sin x and cos x work as thus.

If #f (x) = sin (x),# then #(df)/dx = cos x, (d^2f)/dx^2 = -sin x, (d^3f)/dx^3 = -cos (x), (d^4f)/dx^4 = sin x#

In other words, every 4thmderivative is the original function, as is the eighth, the twelfth, etc. (Note that if we have #sin (u (x)), (df)/dx = (du)/dx cos (u (x)) # instead.

The highest multiple of 4 in our exponent is 84, so we can essentially ignore that part for the derivative.

#(d^86)/dx^86 sin (x) =(d^2)/dx^2 sin (x) = -sin (x)#