It's an atrocious chain rule phenomenon
Let (2x-cos3y)^4sec(ln(1-xy))+sqrt(1+sqrt(y))=pq+r(2x−cos3y)4sec(ln(1−xy))+√1+√y=pq+r
Where p=(2x-cos3y)^4, q=sec(ln(1-xy)), r=sqrt(1+sqrt(y))p=(2x−cos3y)4,q=sec(ln(1−xy)),r=√1+√y
(dp)/(dy) = 4(2x-cos3y)^3 (2 dx/dy+3 sin 3y)dpdy=4(2x−cos3y)3(2dxdy+3sin3y)
(dq)/(dy) = sec(ln(1-xy))tan(ln(1-xy))*((-(dx/dy y + x))/(1-xy))dqdy=sec(ln(1−xy))tan(ln(1−xy))⋅⎛⎜⎝−(dxdyy+x)1−xy⎞⎟⎠
(dq)/(dy) = sec(ln(1-xy))tan(ln(1-xy))*((y dx/dy + x))/(xy-1)dqdy=sec(ln(1−xy))tan(ln(1−xy))⋅(ydxdy+x)xy−1
(dr)/(dy) = (1/(2sqrt(y)))/(2sqrt(1+sqrt(y))) = 1/(4sqrt(y+ysqrt(y)))drdy=12√y2√1+√y=14√y+y√y
Thus d/dy (2x-cos3y)^4sec(ln(1-xy))+sqrt(1+sqrt(y))ddy(2x−cos3y)4sec(ln(1−xy))+√1+√y
=d/dy (pq+r)=ddy(pq+r)
=d/dy (pq) + d/dy r=ddy(pq)+ddyr
=q((dp)/(dy)) + p((dq)/(dy)) + (dr)/(dy)=q(dpdy)+p(dqdy)+drdy
=sec(ln(1-xy))(4(2x-cos3y)^3 (2 dx/dy+3 sin 3y)) + (2x-cos3y)^4(sec(ln(1-xy))tan(ln(1-xy))*((y dx/dy + x))/(xy-1)) + 1/(4sqrt(y+ysqrt(y)))=sec(ln(1−xy))(4(2x−cos3y)3(2dxdy+3sin3y))+(2x−cos3y)4⎛⎜⎝sec(ln(1−xy))tan(ln(1−xy))⋅(ydxdy+x)xy−1⎞⎟⎠+14√y+y√y
