Can one use the Laurent series to find the derivatives at the center of the annulus of convergence?

Question

I was asked to find the Laurent Series of $$f(z)=\frac{1}{z+1}-\frac{1}{z+4}$$ which I found to be, $$f(z)=\sum_{n=0}^\infty (-1)^n\frac{1}{z^{n+1}}-\sum_{n=0}^\infty (-1)^n\frac{z^n}{4^{n+1}}$$ Now how do I find $f^{17}(0)?$

The series above is for $1<|z|<4$.