Chain rule:
2. [tex]\dfrac{\mathrm d}{\mathrm dt}[\cos2t]=-\sin2t\cdot\dfrac{\mathrm d}{\mathrm dt}[2t][/tex]
[tex]=-2\sin2t[/tex]
4. [tex]\dfrac{\mathrm d}{\mathrm dt}[\sin t^2]=\cos t^2\cdot\dfrac{\mathrm d}{\mathrm dt}[t^2}[/tex]
[tex]=2t\cos t^2[/tex]
6. [tex]\dfrac{\mathrm d}{\mathrm dt}[\sin^2t]=2\sin t\cdot\dfrac{\mathrm d}{\mathrm dt}[\sin t][/tex]
[tex]=-2\sin t\cos t[/tex]
[tex]=-\sin2t[/tex]
8. The last function is a bit ambiguous, but I take it to say [tex]f(t)=\sin((2t+1)^2)[/tex], as opposed to [tex]f(t)=\sin^2(2t+1)=(\sin(2t+1))^2[/tex].
[tex]\dfrac{\mathrm d}{\mathrm dt}[\sin(2t+1)^2]=\cos(2t+1)^2\cdot\dfrac{\mathrm d}{\mathrm dt}[(2t+1)^2][/tex]
[tex]=2(2t+1)\cos(2t+1)^2\cdot\dfrac{\mathrm d}{\mathrm dt}[2t+1][/tex]
[tex]=4(2t+1)\cos(2t+1)^2[/tex]
