Derivative Notation $\frac{d^{2}y}{dx^{2}}$, $\frac{d^{3}y}{dx^{3}}$, and $\frac{d^{4}y}{dx^{4}}$

Question

I know that $\frac{dy}{dx}$ is written when you have to find the first derivative of a function. What do $\frac{d^{2}y}{dx^{2}}$, $\frac{d^{3}y}{dx^{3}}$, and $\frac{d^{4}y}{dx^{4}}$ mean though?

Answer 1

They are the second, the third... derivative. For example in physics acceleration is the second derivative of position: $\frac{d^{2}x}{dt^{2}}=a$. They represent the rate of change of the previous derivative. In general a derivative is been written as: $\frac{d^{n}y}{dx^{n}}$ where $n$ is the number of derivative you want to find!