Kinematic equations of SHM in terms of displacement
The equations for the velocity $$(\vecphy{v})$$ and acceleration $$(\vecphy{a})$$ in terms of displacement $$(\vecphy{x})$$ can be derived from the equations in terms of equilibrium or extreme positions and the equation for displacement in terms of time ($$x=x_{0}\sin\omega t$$).
Like all kinematics equations, these can only be applied to linear motion with respect to a particular axis with a specified positive direction. This means that the quantities are represented as scalars, with negative signs to imply motion in the opposite direction.
$$$\begin{align*}v&=\pm\omega\sqrt{x_{0}^{2}x^{2}}\\a&=\omega^{2}x\end{align*}$$$The equation for acceleration is the defining equation for simple harmonic motion (showing the proportionality of $$a$$ and $$x$$).
Some remarks:

$$x$$ is and must always be smaller than $$x_{0}$$ for the velocity to be a real number.

The $$\pm$$ sign corresponds to the fact that the harmonic oscillator periodically changes direction, with velocities of the same magnitude but opposite directions occurring at the same displacement.

The minus sign in $$a=\omega^{2}x$$ stands for the fact that the oscillator strives to restore the equilibrium whenever the displacement is nonzero.
$$x_{0}=$$maximum displacement; $$\omega=$$angular frequency.