What does $\frac{d^2}{dx^2}$ stands for

Question

I would like to know what is $\frac{d^n}{dx^n}$. I think it stands for dervation of $n-$th order but I am not sure.

Answer 1

$$\frac{d^2}{(dx)^2}\,(f(x)) = \frac{d}{dx}\left(\frac d{dx}(f(x))\right)$$

And in general finding $\;\dfrac{d^n}{(dx)^n}\,(f(x))\;$ requires that you successively take the derivative of $f(x)$ $n$ times.

ADDED

You'll also find notation similar to $f'(x), f''(x), f'''(x), f^{(4)}(x), \cdots f^{(n)}(x)$ where for higher order derivatives, we superscript the order and enclose that order in parentheses, to distinguish it from, say, $f^4(x) = (f(x))^4$.

Answer 2

Generally $\frac{d^n}{dx_i^n}f(x_1,x_2,...)$ means the nth derivative of $f$ wrt $x_i$.