An FM signal modulated by $m(t)$ is $x(t)=A_c\cos(\omega_c t+(k_m\int m(\alpha)d\alpha) t)$. This can be expanded as
$$
x(t)=A_c\cos(\omega_c t)sin(\theta_m t)-A_c\sin(\omega_c t)\sin(\theta_m t)
$$
, where $\theta_m = k_m\int m(\alpha)d\alpha$.
For NBFM, $k_m << 1$, so $\theta_m(t) << 1$, so we can approximate
$$
x(t) \approx A_c\cos(\omega_ct)-A_c\theta_m(t)\sin(\omega_ct)
$$
For generating WBFM, frequency multiplier are used to generate NBFM at $n\omega_c$ and $n\Delta f$ (frequency deviation). For example, for 8x multiplication, the signal is passed through square law device 3 times, with each stage having a BPF centered around 2 times center frequency after squaring. For squaring,
$$
x(t)^2 = A_c^2\frac{1}{2}\Big[1+\cos(2\omega_c t)\Big]+A_c^2\theta_m^2(t)\frac{1}{2}\Big[1-\cos(2\omega_ct)\Big] -A_c^2\theta_m(t)\sin(2\omega_ct)
$$
After BPF with center frequency around $2\omega_c$,
$$
\tilde{x_2}(t) = \frac{A_c^2}{2}(1-\theta_m^2(t))\cos(2\omega_ct)-A_c^2\theta_m(t)\sin(2\omega_ct)\\
\approx \frac{A_c^2}{2}\cos(2\omega_ct)-\frac{A_c^2}{2}\Big(2\theta_m(t)\Big)\sin(2\omega_ct)
$$
You can see that not only carrier frequency has doubled but modulation sensitivity has also doubled, resulting in doubling of frequency deviation $\Delta f$.