?DSolve

DSolve[f''[t]==g,f[t],t]

DSolve[{f''[t]==g,f[0]==3,f'[0]==4},f[t],t]

r=DSolve[{f''[t]==g,f[0]==3,f'[0]==4},f[t],t]//Flatten

h[t_]=f[t]/.r

h[0]

D[h[t],t]/.{t->0}