calculus first principles?

Question

I saw a derivation here last week of f '(1/x) from f.p., but now I have lost it

Answer 1

y = (1/x)
y + δy = 1/(x+δx)
δy = 1/(x+δx) - 1/x
δy = [x - (x+δx)]/[x(x+δx)]
δy = -δx / (x(x + δx))
δy/δx = -1/(x(x+δx))

Apply the limit as δx goes to 0
dy/dx = -1/(x(x+δx))
= -1/(x(x+0))
= -1/(x*x)
= -1/x²