Prove by direct computation that $f^{[2]}(z) =\frac{1}{\pi i}\oint_{\partial D_1(0)} \frac{f(w)dw}{(w-z)^3} \text{ for } z \in D_1(0).$

Question

Let $f$ be an analytic function in an open set containing $\overline{D_1}(0)$. Prove by direct computation that $$f^{[2]}(z) =\frac{1}{\pi i}\oint_{\partial D_1(0)} \frac{f(w)dw}{(w-z)^3} \text{ for } z \in D_1(0).$$

I am not really sure how to start this question. Any hints, or partial solutions?