0103 Notations for Derivatives

We will write $$\frac{dy}{dx},y' \text{ and }Dy$$ to all mean the derivative of $y$ with respect to $x$. Only the first one specifies the independent variable $x$. In the other two you can only determine the independent variable from context.
When the independent variable is time $t$ we will usually adopt the physicists' notation $\dot{x}$ for the derivative.

For second derivatives we have $$\frac{d^2y}{dx^2}=y''=D^2y$$ all mean the second derivative of $y$ with respect to $x$. If $x = x(t)$ is a function of time we will also write $\ddot{x}$.

For higher derivatives we will use the notations $$\frac{d^{n}y}{dx^n}=y^{(n)}=D^ny$$ to mean the $n^{th}$ derivative.