What does $\frac{d^6y}{ dx^6}$ mean? The whole question is that 
If $f(x) = -2cos^2x$, then what is $d^6y \over dx^6$ for x = $\pi/4$?
The key here is what does $d^6y \over dx^6$ mean?
I know that $d^6y \over d^6x$ means 6th derivative of y with respect to x, but I've never seen it before.

The symbol "$\frac d{dx}$" is used to indicate a single derivative (with respect to $x$).
We treat repeated application of this operator symbolically as "powers" of the operator (as if it were ordinary multiplication by an ordinary fraction), writing "$\frac{d^n}{dx^n}$" to indicate $n$ successive applications of "$\frac d{dx}$".
The notation is peculiar but wholly accepted as traditional. In particular, one might wonder why "$dx^n$" rather than "$(dx)^n$" in the "denominator"; but evidently the "$d$" isn't regarded as an independent factor, rather "$dx$" is regarded as an atomic term.
One eventually accepts it and gets used to the notation.
