Trigonometric derivatives cycle, as shown. y'=-2sin (2x) y''=-4cos (2x) y'''=8sin (2x) y''''=16cos (2x) So, for the 50th derivative, the co-efficient will be 2^(50). The remainder of 50 divided by 4 is 2, so we know it corresponds with y''. 50th derivative of y=-2^(50)cos(2x) Quadratic Formula. r=(--4pm sqrt((-4)^(2)-4*1*1))/(2*1) r=(4pm sqrt(16-4))/(2) r=2pm sqrt(12) r=2pm 2sqrt(3) Result: 50th derivative of y=-2^(50)cos (2x)