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 correct notation for the sixth derivative is
$$\frac{d^6 y}{dx^6}$$
not $\frac{d^6 y}{d^6x}$. This notation is meant to be suggestive of taking the sixth power of the operator $d/dx$; that is,
$$\frac{d^6 y}{dx^6} = \underbrace{\frac{d}{dx} \cdots \frac{d}{dx}}_{6} y$$
Imagine $dx$ being treated as a single term, of which there are $6$.
