How do you derive the charge on a charging capacitor?

Here is a video with the derivation. It explains the specific configuration above with one resistor and one capacitor. Hope this helps. Are there any other questions you have?

Thank you. This is very helpful. I have another related question. What if the circuit has no battery? The capacitor has a charge Q and is discharging when a switch is closed. How do we derive the equation for the charge on the discharging capacitor?

This is another video for a charge on a discharging capacitor. Let me know if you have any other questions.

Thanks, that helped me a lot.