Putnam 1992 A4

Let $f$ be an infinitely differentiable real-valued function defined on the real numbers. If $$f\left( \frac{1}{n} \right) = \frac{n^2}{n^2 + 1}, n = 1, 2, 3, \dots,$$ compute the values of the derivatives $f^{(k)}(0), k = 1, 2, 3, \dots$.