Let $f(x) = arctan (x)$. Use the derivative approximation: $f'(x) = \frac{8f(x+h) - 8f(x-h) - f(x+2h) +f(x-2h)}{12h} $ to approximate $f'(\frac14\pi)$ using $h^-1$ = 2, 4, 8 . Try to take h small enough that the rounding error effect begins to dominate the mathematical error. For what value of h does this begin to occur? (You may have to restrict yourself to working in single precision.)
I have no idea how to even begin...
Partial answer
To get an idea how good the approximation is, you can calculate the result for the special functions $f(x)=1,x,x^2,x^3\cdots$. Here, we get the exact result for $f(x)=1,x,x^2,x^3,x^4$, so the formula is exact upto degree $4$ (Polynomials upto degree $4$ are differentiated exactly by the formula)
For $f(x)=x^5$, you get $f'(x)=5x^4-4h^4$, so you have an error of $4h^4$