Find the Fourier transform of the function

$$f(x)=\frac{1}{1+x^{2}}, \quad x \in \mathbb{R}$$

using an appropriate contour integration. Hence find the Fourier transform of its derivative, $f^{\prime}(x)$, and evaluate the integral

$$I=\int_{-\infty}^{\infty} \frac{4 x^{2}}{\left(1+x^{2}\right)^{4}} d x$$