Find `(dx)/(dt)` (derivative) of w.r.t. t).
(i) `x=(t^(2)+1)^(3)` (ii) `x=sqrt(t)^(3)-3` (iii) `x=sin2t` (iv) `x=cos(2t+4)` (v)`x=sin^(3)t` (vi)`x=cos^(3)t`

Answer

(i) ` 6t(t^(2)+1)^(2)` (ii)`(3t^(2))/(2)(t^(3)-3)^(-1//2)` (iii) `2cos 2t` (iv) `-2sin(2t+4)`(v)`3sin^(2)tcos t` (vi) `-3cos^(2) t sin t `

Explanation

(i) `(dx)/(dt)=3(t^(2)+1)^(2)(2t)=6t(t^(2)+1)^(2)`
(ii)`(dx)/(dt)=(1)/(2)(t^(3)-3)^(-1/2)(3t^(2))=(3t^(2))/(2)(t^(3)-3)^(-1/2)`
(iii) `(dx)/(dt)=2 cos 2t`
(iv)`(dx)/(dt)=-sin(2t+4).2 =-2sin (2t+4)`
(v) `(dx)/(dt)=3sin ^(2)(cos t)=3 sin^(2) .cost`
vi) `(dx)/(dy)=3 cos^(2) t(-sin t)=-3 cos^(2) t sin t`