Let
y=(cos(2x^4)-1)/(x^7)
Let
u=cos(2x^4)-1
v=x^7
y=u/v
By quotient rule,
dy/dx=(v(du)/dx-u(dv)/dx)/v^2
u=cos(2x^4)-1
Let
t=x^4
u=cos2t-1
wkt
1-cos2t=2sin^2t
u=-(1-cos2t)
u=-2sin^2t
By chain rule
(du)/dx=(du)/dt(dt)/dx
u=-2sin^2t
sin^2t=(sint)^2
Let
p=sint
(sint)^2=p^2
u=-2p
By chain rule
(du)/dt=-2(dp)/dt
p=sint
(dp)/dt=cost
(du)/dt=-2cost
-2cost=-2cosx^4
(du)/dt=-2cosx^4
(du)/dx=(du)/dt(dt)/dx
t=x^4
(dt)/dx=4x^3
(du)/dx=-2cosx^4(4x^3)
(du)/dx=-8x^3cosx^4
v=x^7
(dv)/dx=7x^6
dy/dx=(v(du)/dx-u(dv)/dx)/v^2
u=-2sin^2t
u=-2sin^2x^4
v=x^7
(du)/dx=-8x^3cosx^4
(dv)/dx=7x^6
dy/dx=(x^7(-8x^3cosx^4)-(-2sin^2x^4)(7x^6))/(x^7)^2
dy/dx=(-8x^10cosx^4+14x^6sin^2x^4)/x^14
dy/dx=-8/x^4cosx^4+14/x^8sin^2x^4
