What is the change in voltage across an inductor, or change in current through a capacitor?
Your circuit diagram shows an ideal voltage source with 1 V output, from \$t=-\infty\$ to \$t=\infty\$. Since the voltage is constantly 1 V, \$\frac{dV}{dt}=0\$.

If you want a nonzero \$\frac{dV}{dt}\$ you should include some mechanism in the circuit to change the votlage, or use a source with a varying output voltage.

In early classes, this is often done with switches. In the real world it might also be done with transistors or various other mechanisms.


  What if there was a switch in this circuit, so there was 0v and then the voltage changed to 1v?


Then the rate of change of the capacitor voltage would depend on parasitic resistances you haven't shown in the circuit. The voltage source has an effective resistance of some kind, the capacitor has an effective series resistance (ESR), and the wires have resistance. 

If there's no discrete resistor added to the circuit, these unknown parasitics will determine the rate the capacitor charges at.

If these parasitic resitances are very low, then the parasitic inductance of the wires and of the capacitor could also play a role.