Show that the following functions are flows on the spaces indicated. Find the vector field for each flow $$\phi_t(x)=\frac{x+\tanh t}{1+ x \tanh t}, x \in [-1,1]$$
Solution so far
So $\phi_0(x)= \frac{x+\tanh 0}{1+ x \tanh 0}=x$ And $$\phi_t \circ \phi_s=\frac{\phi_s(x)+\tanh t}{1+\phi_s(x) \tanh t}=\frac{\frac{x+\tanh s}{1+ x \tanh s}+\tanh t}{1+ \frac{x+\tanh s}{1+ x \tanh s}\tanh t}=\frac{x \cosh(s+t)+\sinh(s+t)}{x\sinh(s+t)+\cosh(s+t)}=\frac{x+\tanh(t+s)}{1+x \tanh(t+s)}=\phi_{t+s}$$ Is what I have so far correct? And can anyone help me with finding the vector field for the flow?
All is correct, and the vector field is given by $$ f(x)=\frac{\partial}{\partial t}\phi_t(x)\Big|_{t=0}. $$ PS: It should be something like "Show that the following functions form a flow on the space indicated. Find the vector field for the flow".