Band edge filter for raised cosine impulse
It looks like the filter you want is indeed $-ith(t)$.  Here is some Octave code to get a visualization in the frequency domain:

t = [-1:0.01:1];
h = 0.5*(1+cos(pi*t));
hd = -i*t*0.5.*(1+cos(pi*t));
H=fftshift(fft(h,512));
HD=fftshift(fft(hd,512));
v = [-256:255];
plot(v, 20*log10(abs(H/512)), v, 20*log10(abs(HD/512)))




BTW, when plotting $h(t)$ in the time domain, your provided $h(t)$ appears to be normalized, spanning from $t = -1$ to $t = 1$.  So, I'm guessing your normalized symbol period is $T = 1$ with some ISI, or $T = 2$ with no ISI.