I’m trying to find the arc length of a curve and I only wanted to know if step 1 is correct so far then if I have trouble I’ll ask for help with my solution method
Find the derivate
$$r ( t ) = \langle \sin ( t^2) , \cos ( t^2 ) , t^3 \rangle $$
$$r' ( t ) = \langle 2 t \cos ( t^2 ) , - 2 t \sin ( t^2 ) , 3 t^2 \rangle$$
With your updated question text, your differentiations are correct. For example with the first one, you can use the chain rule to get the derivative of $\sin$ is $\cos$ and that $t^2$ is $2t$ to put it together to get that
$$\begin{equation}\begin{aligned} \frac{d(\sin(t^2))}{dt} & = \left(\frac{d(\sin(t^2))}{d(t^2)}\right)\left(\frac{d(t^2)}{dt}\right) \\ & = \cos(t^2)(2t) \end{aligned}\end{equation}\tag{1}\label{eq1A}$$
