The displacement x of a particle moving along x-axis at time t is given by $x^2 =2t^2 +6t$. What is the velocity at any time t?

Dear Student,
                      As given in the question, $x^2=2t^2+6t$
                    So, we can write the displacement, $x = \sqrt{2t^2+6t}$             ........(1)
                So, the velocity at time 't',  $$\begin{gathered} v = \frac{dx}{dt} = \frac{d}{dt}\left(\sqrt{2t^2+6t}\right) \\ = \frac{1}{2}\frac{\left[4t+6\right]}{\sqrt{2t^2+6t}} \\ =\frac{2t+3}{x} [ u\sin g equation (1\left)\right] \end{gathered}$$
Hence the correct option is (4)


Regards