Stupid math question

I took my calc 1 midterm on friday, and i've had this one question stuck in my head since.

"Find the derivative of 1/x using increments"

I know the answer is dy/dx = -1/x^2 from the power rule. But I couldn't figure out how to get from dy/dx = [(1/(x+dx))-(1/x)]/dx to that.

Surely use partial fractions, no idea what increments are, sounds like shitty high school maths.

1/(x+dx) - 1/x = -dx/x(x+dx)

so dy/dx = (-dx/x(x+dx))/dx = -1/x(x+dx) ---> -1/x^2

Ah, that makes since, thanks.

*sense