According to the definition, there is (sinx)'=lim[sin(x+△x)-sinx]/(△x), where △x→0, and expand sin(x+△x)-sinx, which is sinxcos△x+cosxsin△x-sinx, because △x→0. So (sinx)'=lim(cosxsin△x)/△x, an important limit must be used here. When △x→0, lim(sin△x)/△x= 1, so (sinx)'=cosx.

Similarly, (cosx)'=lim[cos(x+△x)-cosx]/△x, where △x→0. At this time, cos (x+△ x)-cosx = cosx cos △ x-cosx →-sinxsin △ x, (cosx).

(lnx)' = lim [ln (x+△ x)-lnx]/△ x，△ x→ 0。 Ln (x+△ x)-lnx = ln (1+△ x/x), there is also a limit: when t→0, ln (.

The formula of bottoming is logaX=lnX/lna=(loga e)lnX, and we get (lnX)'= 1/X, so [logax]' = [(logae) lnx]' = (logae)/x.

The derivation of these formulas requires some important limits that are not mentioned in middle school textbooks, so the textbooks directly write out the results without deriving formulas. So much for my answer. If you don't understand anything, please continue the discussion.