$$
u(t) = \begin{cases}
1 & t \ge 0 \\
0 & t < 0
\end{cases}\space ;\space
u(t -t_0) = \begin{cases}
1 & t \ge t_0 \\
0 & t < t_0
\end{cases}\space;\space
u(at) = \begin{cases}
u(t) & a > 0 \\
u(-t) & otherwise
\end{cases}
$$
$$
\implies\space \frac{d(u(t))}{dt}=\delta(t)=impluse\space function \space;\space \int_{-\infty}^{t} u(t) dt=tu(t) = r(t) =ramp\space function
$$
$$
\delta(t) = \begin{cases}
0 & t \ne 0 \\
\infty & t = 0
\end{cases}\space ;\space
\delta(t -t_0) = \begin{cases}
0 & t \ne t_0 \\
\infty & t = t_0
\end{cases}\space;\space
\delta(at)=\frac{1}{\lvert a \rvert}\delta(t)
\implies \delta(t) = \frac{d(u(t))}{dt} \space;\space \int_{-\infty}^{\infty} \delta(t) dt=1
$$
$$
r(t)=\begin{cases}
t & t \ge 0 \\
0 & t < 0
\end{cases}=tu(t)\space ;\space
r(t-t_0)=(t-t_0)u(t-t_0)\space ; \space r(at)=ar(t)
$$
$$
\implies \frac{d(r(t))}{dt}=u(t) \space;\space \int_{-\infty}^{t} r(t) dt=\frac{t^2}{2} u(t)
$$
$$
x(t) = e^{at}\space;\space x(t-t_0) = e^{a(t-t_0)} = e^{-at_0}e^{at} \space ; \space x(bt) = e^{abt} \implies \frac{d(x(t))}{dt} = a e^{at}; \int x(t) dt = \frac{1}{a}e^{at}
$$
$$
x(t) = A \sin(\omega_0 t)\space ; \space x(t-t_0) = A \sin(\omega_0 (t-t_0))\space;\space\sin(\omega_0 t) = \frac{e^{j\omega_0 t} - e^{-j\omega_0 t}}{2j}
$$
$$
\implies \frac{d(x(t))}{dt} = A\omega_0 \cos(\omega_0 t) ; \quad \int x(t) dt = -\frac{A}{\omega_0} \cos(\omega_0 t) +C
$$