About FM signal's proving My text book (Communication Systems analysis and design P.334, 335) have a FM signal is 


  $s_{FM}(t) = A cos[2\pi[f_c+ks(t)]t+\phi]$ where k is a constant and $\phi$ is the arbitrary phase angles


then the textbook suggest


  while in FM, instantaneous frequency minus carrier frequency is a linear function of s(t)


So, I try to prove the $s_{FM}(t)$ is a FM signal

The instantaneous frequency is $f(t)= \frac {d\psi}{dt}=2\pi f_c+2 \pi s'(t) t +2 \pi s(t)$

and the instantaneous carrier ferquency is $f_{carrier} (t)=\frac{d\psi_{carrier}}{dt}=\frac {d}{dt} [2 \pi f_c t + \theta]=2 \pi f_c$. Assume the carrier signal is $Asin(2 \pi f_c t + \theta)$

So, $f(t)-f_{carrier}(t)=2 \pi s'(t) t +2 \pi s(t)$.

And my problem is why the text book suggest that signal is a FM signal? Clearly, I cannot prove the signal is linear.
As you have correctly derived, the difference of carrier and instantaneous frequency is

$$
g(s(t)) = \omega(t) - \omega_\mathrm{carrier} = 2\pi k s(t) + 2\pi k s(t)'t
$$

I will omit $t$ for convenience. If $g(s)$ is a linear function of $s$ it must fulfill the following condition:

$$
g(c(a + b)) = cg(a) + cg(b)
$$
So here:
$$
g(c(a+b)) = 2\pi k c(a + b) + 2\pi k \left[\frac{d}{dt}(c(a + b))\right]t\\
= 2\pi k c a + 2\pi k c b + 2\pi k c a' t + 2\pi k c b' t\\
= c(2\pi k a + 2\pi k a' t) + c(2\pi k b + 2\pi k b' t)\\
= cg(a) + cg(b)
$$
Thus, $g(s(t))$ is indeed a linear function of $s(t)$.