Find the first derivatives
(i) If `y=x^(-1//3)," find "(dy)/(dx)`
(ii) If `x=t^(6)," find "(dx)/(dt)`
(iii) If `z=u^(3)," find "(dz)/(du)`
(iv) If `f=r^(2)+r^(1//3)," find "(df)/(dr)`
(v) If `x=2t^(3)-3t^(2)+5t-6," find "(dx)/(dt)`
(vi) If `y=3x^(2)`
(vii) If `y=xlnx`
(viii) If `y=xsinx`
(ix) If `y=sinx cosx`
(x) If `y=sin^(2)x+cos^(2)x`
(xi) If `y=cos2x`
(xii) If `v=(4)/(3)pir^(3)," find "(dv)/(dr)`
(xiii) If `s=4pir^(2)," find "(ds)/(dr)`

Answer

(i)`-(1)/(3)x^(-4//3)`
(ii)`6t^(5)`
(iii)`3u^(3)`
(iv)`2r+(1)/(3)r^(-2//3)`
(v)`6t^(2)-6t+5`
(vi)`6x`
(vii)`1+lnx`
(viii)`xcosx+sinx`
(ix)`cos^(2)x-sin^(2)x`
(x)0
(xi)`-2sin2x`
(xii)`4pir^(2)`
(xiii)`8pir`