Question

I would like some help with those 4 please. Calculus! Show your work please! Thanks!

Answer 1

Chain rule:

2.
`(\mathrm d)/(\mathrm dt)[\cos2t]=-\sin2t\cdot(\mathrm d)/(\mathrm dt)[2t]`

`=-2\sin2t`

4.
`(\mathrm d)/(\mathrm dt)[\sin t^2]=\cos t^2\cdot(\mathrm d)/(\mathrm dt)[t^2}`

`=2t\cos t^2`

6.
`(\mathrm d)/(\mathrm dt)[\sin^2t]=2\sin t\cdot(\mathrm d)/(\mathrm dt)[\sin t]`

`=-2\sin t\cos t`

`=-\sin2t`

8. The last function is a bit ambiguous, but I take it to say
`f(t)=\sin((2t+1)^2)`, as opposed to
`f(t)=\sin^2(2t+1)=(\sin(2t+1))^2`.

`(\mathrm d)/(\mathrm dt)[\sin(2t+1)^2]=\cos(2t+1)^2\cdot(\mathrm d)/(\mathrm dt)[(2t+1)^2]`

`=2(2t+1)\cos(2t+1)^2\cdot(\mathrm d)/(\mathrm dt)[2t+1]`

`=4(2t+1)\cos(2t+1)^2`