Well for now, just remember a derivative describes a function's rate of change.
Without overcomplicating things, I'll just say that when I say "differentiate with respect to time," that means: (1) time is the independent variable (the x-axis), (2) voltage is the dependent variable (the y-axis), and (3) the derivative is the slope. Technically, you can also "differentiate with respect to voltage," but this would be like saying 'hours per mile' instead of 'miles per hour.' ...And only weirdos differentiate the independent variable. :lol:
For sine waves, we've memorized from calculus class that the derivative of sine is cosine. If you plot sine and cosine on a graph, you see that they're identical, except cosine has a 90 degree "head start" on sine. This is the phase shift.
For now, don't worry about how inductors and capacitors do the differention (I don't fully understand Maxwell's equations myself).